laventua

lavennun (first-person indicative present singular)
en lavennu (first-person indicative negative present singular)
olen laventunut (first-person indicative perfect singular)
en ole laventunut (first-person indicative negative perfect singular)
lavennut (indicative present second-person singular)
et lavennu (indicative negative present second-person singular)
olet laventunut (indicative perfect second-person singular)
et ole laventunut (indicative negative perfect second-person singular)
laventuu (indicative present singular third-person)
ei lavennu (indicative negative present singular third-person)
on laventunut (indicative perfect singular third-person)
ei ole laventunut (indicative negative perfect singular third-person)
lavennumme (first-person indicative plural present)
emme lavennu (first-person indicative negative plural present)
olemme laventuneet (first-person indicative perfect plural)
emme ole laventuneet (first-person indicative negative perfect plural)
lavennutte (indicative plural present second-person)
ette lavennu (indicative negative plural present second-person)
olette laventuneet (indicative perfect plural second-person)
ette ole laventuneet (indicative negative perfect plural second-person)
laventuvat (indicative plural present third-person)
eivät lavennu (indicative negative plural present third-person)
ovat laventuneet (indicative perfect plural third-person)
eivät ole laventuneet (indicative negative perfect plural third-person)
lavennutaan (indicative passive present)
ei lavennuta (indicative negative passive present)
on lavennuttu (indicative passive perfect)
ei ole lavennuttu (indicative negative passive perfect)
lavennuin (first-person indicative past singular)
en laventunut (first-person indicative negative past singular)
olin laventunut (first-person indicative pluperfect singular)
en ollut laventunut (first-person indicative negative pluperfect singular)
lavennuit (indicative past second-person singular)
et laventunut (indicative negative past second-person singular)
olit laventunut (indicative pluperfect second-person singular)
et ollut laventunut (indicative negative pluperfect second-person singular)
laventui (indicative past singular third-person)
ei laventunut (indicative negative past singular third-person)
oli laventunut (indicative pluperfect singular third-person)
ei ollut laventunut (indicative negative pluperfect singular third-person)
lavennuimme (first-person indicative past plural)
emme laventuneet (first-person indicative negative past plural)
olimme laventuneet (first-person indicative pluperfect plural)
emme olleet laventuneet (first-person indicative negative pluperfect plural)
lavennuitte (indicative past plural second-person)
ette laventuneet (indicative negative past plural second-person)
olitte laventuneet (indicative pluperfect plural second-person)
ette olleet laventuneet (indicative negative pluperfect plural second-person)
laventuivat (indicative past plural third-person)
eivät laventuneet (indicative negative past plural third-person)
olivat laventuneet (indicative pluperfect plural third-person)
eivät olleet laventuneet (indicative negative pluperfect plural third-person)
lavennuttiin (indicative passive past)
ei lavennuttu (indicative negative passive past)
oli lavennuttu (indicative passive pluperfect)
ei ollut lavennuttu (indicative negative passive pluperfect)
laventuisin (conditional first-person present singular)
en laventuisi (conditional first-person negative present singular)
olisin laventunut (conditional first-person perfect singular)
en olisi laventunut (conditional first-person negative perfect singular)
laventuisit (conditional present second-person singular)
et laventuisi (conditional negative present second-person singular)
olisit laventunut (conditional perfect second-person singular)
et olisi laventunut (conditional negative perfect second-person singular)
laventuisi (conditional present singular third-person)
ei laventuisi (conditional negative present singular third-person)
olisi laventunut (conditional perfect singular third-person)
ei olisi laventunut (conditional negative perfect singular third-person)
laventuisimme (conditional first-person plural present)
emme laventuisi (conditional first-person negative plural present)
olisimme laventuneet (conditional first-person perfect plural)
emme olisi laventuneet (conditional first-person negative perfect plural)
laventuisitte (conditional plural present second-person)
ette laventuisi (conditional negative plural present second-person)
olisitte laventuneet (conditional perfect plural second-person)
ette olisi laventuneet (conditional negative perfect plural second-person)
laventuisivat (conditional plural present third-person)
eivät laventuisi (conditional negative plural present third-person)
olisivat laventuneet (conditional perfect plural third-person)
eivät olisi laventuneet (conditional negative perfect plural third-person)
lavennuttaisiin (conditional passive present)
ei lavennuttaisi (conditional negative passive present)
olisi lavennuttu (conditional passive perfect)
ei olisi lavennuttu (conditional negative passive perfect)
lavennu (imperative present second-person singular)
älä lavennu (imperative negative present second-person singular)
laventukoon (imperative present singular third-person)
älköön laventuko (imperative negative present singular third-person)
olkoon laventunut (imperative perfect singular third-person)
älköön olko laventunut (imperative negative perfect singular third-person)
laventukaamme (first-person imperative plural present)
älkäämme laventuko (first-person imperative negative plural present)
laventukaa (imperative plural present second-person)
älkää laventuko (imperative negative plural present second-person)
laventukoot (imperative plural present third-person)
älkööt laventuko (imperative negative plural present third-person)
olkoot laventuneet (imperative perfect plural third-person)
älkööt olko laventuneet (imperative negative perfect plural third-person)
lavennuttakoon (imperative passive present)
älköön lavennuttako (imperative negative passive present)
olkoon lavennuttu (imperative passive perfect)
älköön olko lavennuttu (imperative negative passive perfect)
laventunen (first-person potential present singular)
en laventune (first-person negative potential present singular)
lienen laventunut (first-person perfect potential singular)
en liene laventunut (first-person negative perfect potential singular)
laventunet (potential present second-person singular)
et laventune (negative potential present second-person singular)
lienet laventunut (perfect potential second-person singular)
et liene laventunut (negative perfect potential second-person singular)
laventunee (potential present singular third-person)
ei laventune (negative potential present singular third-person)
lienee laventunut (perfect potential singular third-person)
ei liene laventunut (negative perfect potential singular third-person)
laventunemme (first-person plural potential present)
emme laventune (first-person negative plural potential present)
lienemme laventuneet (first-person perfect plural potential)
emme liene laventuneet (first-person negative perfect plural potential)
laventunette (plural potential present second-person)
ette laventune (negative plural potential present second-person)
lienette laventuneet (perfect plural potential second-person)
ette liene laventuneet (negative perfect plural potential second-person)
laventunevat (plural potential present third-person)
eivät laventune (negative plural potential present third-person)
lienevät laventuneet (perfect plural potential third-person)
eivät liene laventuneet (negative perfect plural potential third-person)
lavennuttaneen (passive potential present)
ei lavennuttane (negative passive potential present)
lienee lavennuttu (passive perfect potential)
ei liene lavennuttu (negative passive perfect potential)
laventua (infinitive infinitive-i)
laventuva (active participle present)
lavennuttava (participle passive present)
laventuakseni (first-person possessive singular)
laventuaksemme (first-person plural possessive)
laventuaksesi (possessive second-person singular)
laventuaksenne (plural possessive second-person)
laventuakseen (possessive third-person)
laventuaksensa (possessive third-person)
laventunut (infinitive-i-long past)
lavennuttu (infinitive-i-long past)
laventuessa (inessive second-person)
lavennuttaessa (inessive second-person)
laventuma (agent)
laventuessani (first-person possessive singular)
laventuessamme (first-person plural possessive)
laventuessasi (possessive second-person singular)
laventuessanne (plural possessive second-person)
laventuessaan (possessive third-person)
laventuessansa (possessive third-person)
laventumaton (inessive negative second-person)
laventuen (instructive second-person)
laventumassa (inessive third-person)
laventumasta (elative third-person)
laventumaan (illative third-person)
laventumalla (adessive third-person)
laventumatta (abessive third-person)
laventuman (instructive third-person)
lavennuttaman (instructive third-person)
laventuminen (fourth-person noun-from-verb)
laventumaisillani (first-person possessive singular)
laventumaisillamme (first-person plural possessive)
laventumaisillasi (possessive second-person singular)
laventumaisillanne (plural possessive second-person)
laventumaisillaan (possessive third-person)
laventumaisillansa (possessive third-person)

[ˈlɑ̝ʋe̞n̪ˌt̪uɑ̝(ʔ)]


lieventyä

lievennyn (first-person indicative present singular)
en lievenny (first-person indicative negative present singular)
olen lieventynyt (first-person indicative perfect singular)
en ole lieventynyt (first-person indicative negative perfect singular)
lievennyt (indicative present second-person singular)
et lievenny (indicative negative present second-person singular)
olet lieventynyt (indicative perfect second-person singular)
et ole lieventynyt (indicative negative perfect second-person singular)
lieventyy (indicative present singular third-person)
ei lievenny (indicative negative present singular third-person)
on lieventynyt (indicative perfect singular third-person)
ei ole lieventynyt (indicative negative perfect singular third-person)
lievennymme (first-person indicative plural present)
emme lievenny (first-person indicative negative plural present)
olemme lieventyneet (first-person indicative perfect plural)
emme ole lieventyneet (first-person indicative negative perfect plural)
lievennytte (indicative plural present second-person)
ette lievenny (indicative negative plural present second-person)
olette lieventyneet (indicative perfect plural second-person)
ette ole lieventyneet (indicative negative perfect plural second-person)
lieventyvät (indicative plural present third-person)
eivät lievenny (indicative negative plural present third-person)
ovat lieventyneet (indicative perfect plural third-person)
eivät ole lieventyneet (indicative negative perfect plural third-person)
lievennytään (indicative passive present)
ei lievennytä (indicative negative passive present)
on lievennytty (indicative passive perfect)
ei ole lievennytty (indicative negative passive perfect)
lievennyin (first-person indicative past singular)
en lieventynyt (first-person indicative negative past singular)
olin lieventynyt (first-person indicative pluperfect singular)
en ollut lieventynyt (first-person indicative negative pluperfect singular)
lievennyit (indicative past second-person singular)
et lieventynyt (indicative negative past second-person singular)
olit lieventynyt (indicative pluperfect second-person singular)
et ollut lieventynyt (indicative negative pluperfect second-person singular)
lieventyi (indicative past singular third-person)
ei lieventynyt (indicative negative past singular third-person)
oli lieventynyt (indicative pluperfect singular third-person)
ei ollut lieventynyt (indicative negative pluperfect singular third-person)
lievennyimme (first-person indicative past plural)
emme lieventyneet (first-person indicative negative past plural)
olimme lieventyneet (first-person indicative pluperfect plural)
emme olleet lieventyneet (first-person indicative negative pluperfect plural)
lievennyitte (indicative past plural second-person)
ette lieventyneet (indicative negative past plural second-person)
olitte lieventyneet (indicative pluperfect plural second-person)
ette olleet lieventyneet (indicative negative pluperfect plural second-person)
lieventyivät (indicative past plural third-person)
eivät lieventyneet (indicative negative past plural third-person)
olivat lieventyneet (indicative pluperfect plural third-person)
eivät olleet lieventyneet (indicative negative pluperfect plural third-person)
lievennyttiin (indicative passive past)
ei lievennytty (indicative negative passive past)
oli lievennytty (indicative passive pluperfect)
ei ollut lievennytty (indicative negative passive pluperfect)
lieventyisin (conditional first-person present singular)
en lieventyisi (conditional first-person negative present singular)
olisin lieventynyt (conditional first-person perfect singular)
en olisi lieventynyt (conditional first-person negative perfect singular)
lieventyisit (conditional present second-person singular)
et lieventyisi (conditional negative present second-person singular)
olisit lieventynyt (conditional perfect second-person singular)
et olisi lieventynyt (conditional negative perfect second-person singular)
lieventyisi (conditional present singular third-person)
ei lieventyisi (conditional negative present singular third-person)
olisi lieventynyt (conditional perfect singular third-person)
ei olisi lieventynyt (conditional negative perfect singular third-person)
lieventyisimme (conditional first-person plural present)
emme lieventyisi (conditional first-person negative plural present)
olisimme lieventyneet (conditional first-person perfect plural)
emme olisi lieventyneet (conditional first-person negative perfect plural)
lieventyisitte (conditional plural present second-person)
ette lieventyisi (conditional negative plural present second-person)
olisitte lieventyneet (conditional perfect plural second-person)
ette olisi lieventyneet (conditional negative perfect plural second-person)
lieventyisivät (conditional plural present third-person)
eivät lieventyisi (conditional negative plural present third-person)
olisivat lieventyneet (conditional perfect plural third-person)
eivät olisi lieventyneet (conditional negative perfect plural third-person)
lievennyttäisiin (conditional passive present)
ei lievennyttäisi (conditional negative passive present)
olisi lievennytty (conditional passive perfect)
ei olisi lievennytty (conditional negative passive perfect)
lievenny (imperative present second-person singular)
älä lievenny (imperative negative present second-person singular)
lieventyköön (imperative present singular third-person)
älköön lieventykö (imperative negative present singular third-person)
olkoon lieventynyt (imperative perfect singular third-person)
älköön olko lieventynyt (imperative negative perfect singular third-person)
lieventykäämme (first-person imperative plural present)
älkäämme lieventykö (first-person imperative negative plural present)
lieventykää (imperative plural present second-person)
älkää lieventykö (imperative negative plural present second-person)
lieventykööt (imperative plural present third-person)
älkööt lieventykö (imperative negative plural present third-person)
olkoot lieventyneet (imperative perfect plural third-person)
älkööt olko lieventyneet (imperative negative perfect plural third-person)
lievennyttäköön (imperative passive present)
älköön lievennyttäkö (imperative negative passive present)
olkoon lievennytty (imperative passive perfect)
älköön olko lievennytty (imperative negative passive perfect)
lieventynen (first-person potential present singular)
en lieventyne (first-person negative potential present singular)
lienen lieventynyt (first-person perfect potential singular)
en liene lieventynyt (first-person negative perfect potential singular)
lieventynet (potential present second-person singular)
et lieventyne (negative potential present second-person singular)
lienet lieventynyt (perfect potential second-person singular)
et liene lieventynyt (negative perfect potential second-person singular)
lieventynee (potential present singular third-person)
ei lieventyne (negative potential present singular third-person)
lienee lieventynyt (perfect potential singular third-person)
ei liene lieventynyt (negative perfect potential singular third-person)
lieventynemme (first-person plural potential present)
emme lieventyne (first-person negative plural potential present)
lienemme lieventyneet (first-person perfect plural potential)
emme liene lieventyneet (first-person negative perfect plural potential)
lieventynette (plural potential present second-person)
ette lieventyne (negative plural potential present second-person)
lienette lieventyneet (perfect plural potential second-person)
ette liene lieventyneet (negative perfect plural potential second-person)
lieventynevät (plural potential present third-person)
eivät lieventyne (negative plural potential present third-person)
lienevät lieventyneet (perfect plural potential third-person)
eivät liene lieventyneet (negative perfect plural potential third-person)
lievennyttäneen (passive potential present)
ei lievennyttäne (negative passive potential present)
lienee lievennytty (passive perfect potential)
ei liene lievennytty (negative passive perfect potential)
lieventyä (infinitive infinitive-i)
lieventyvä (active participle present)
lievennyttävä (participle passive present)
lieventyäkseni (first-person possessive singular)
lieventyäksemme (first-person plural possessive)
lieventyäksesi (possessive second-person singular)
lieventyäksenne (plural possessive second-person)
lieventyäkseen (possessive third-person)
lieventyäksensä (possessive third-person)
lieventynyt (infinitive-i-long past)
lievennytty (infinitive-i-long past)
lieventyessä (inessive second-person)
lievennyttäessä (inessive second-person)
lieventymä (agent)
lieventyessäni (first-person possessive singular)
lieventyessämme (first-person plural possessive)
lieventyessäsi (possessive second-person singular)
lieventyessänne (plural possessive second-person)
lieventyessään (possessive third-person)
lieventyessänsä (possessive third-person)
lieventymätön (inessive negative second-person)
lieventyen (instructive second-person)
lieventymässä (inessive third-person)
lieventymästä (elative third-person)
lieventymään (illative third-person)
lieventymällä (adessive third-person)
lieventymättä (abessive third-person)
lieventymän (instructive third-person)
lievennyttämän (instructive third-person)
lieventyminen (fourth-person noun-from-verb)
lieventymäisilläni (first-person possessive singular)
lieventymäisillämme (first-person plural possessive)
lieventymäisilläsi (possessive second-person singular)
lieventymäisillänne (plural possessive second-person)
lieventymäisillään (possessive third-person)
lieventymäisillänsä (possessive third-person)


left ideal

If I is a left ideal of R then rI⊆I for any r isin R.

(algebra) A subset of a ring which is closed under left-multiplication by any element of the ring.


right ideal

If I is a right ideal of R then Ir⊆I for any r isin R.

(algebra) A subset of a ring which is closed under right-multiplication by any element of the ring.


two-sided ideal

en-noun

two-sided ideal (plural two-sided ideals)

(algebra) A subset of a ring which is both a left ideal and a right ideal. Commonly referred to simply as an ideal, but referred to as two-sided ideal for emphasis.


principal ideal

principal ideal (plural principal ideals)

(algebra) An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a.

two-sided principal ideal

smallest ideal that contains a given element


principal ideal domain

principal ideal domain (plural principal ideal domains)

(algebra) An integral domain in which every ideal is a principal ideal.


izomorfio

eo-head

izomorfio (accusative singular izomorfion, plural izomorfioj, accusative plural izomorfiojn)

izomorfion (accusative singular)
izomorfioj (plural)
izomorfiojn (accusative plural)


semisimple

LL-Q1860 (eng)-Flame, not lame-semisimple.wav

(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.

(category theory, most generally, of an abelian category) Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects.

(module theory, of a module) In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules.

(ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.

(of a ring, somewhat proscribed) Semiprimitive: having trivial Jacobson radical.

(linear algebra, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.

(Lie theory, of a Lie algebra) Being a direct sum of simple Lie algebras.

linear representation
linear representation

simple representation
simple representation

irreducible representation
irreducible representation

(representation theory, of a linear representation of a group or algebra) Being a direct sum of simple representations (also known as irreducible representations).

(group theory, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial.


associator

(algebra) A multilinear map, given by [x, y, z] = xy(z) − x(yz), that measures the degree of nonassociativity of a ring or algebra.

Assoziator

multilinear map measuring the degree of nonassociativity


PID

principal ideal domain
principal ideal domain#English

(algebra) Initialism of principal ideal domain.


divisibility sequence

en-noun

divisibility sequence (plural divisibility sequences)

If #92;mathcalC#61;(C#95;n)#95;#123;n#92;ge 1#125; is any divisibility sequence, one says that a prime p is a primitive divisor of C#95;n if p#92;vertC#95;n but p#92;nmidC#95;1C#95;2C#95;#123;n-1#125;. Primitive divisors of certain divisibility sequences were studied by Zsigmondy [37] in the 19ᵗʰ century.

2012, P. Ingram, J. H. Silverman, “Primitive Divisors in Elliptic Divisibility Sequences”, in Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth Ribet, John Tate, editors, Number Theory, Analysis and Geometry, Springer,, page 244:

These divisibility sequences satisfy the same recurrence relations as the polynomials #92;phi#95;n and #92;psi²#95;n (see Appendix C).

2013, Graham Everest, Thomas Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, page 138:

Moreover, in order to keep a divisibility sequence normalized, we can assume without loss of generality that a#95;0 = 0 and a#95;1#61;1.

2021, Masum Billal, Samin Riasat, Integer Sequences, Springer, page 59:

(number theory, algebra) Any sequence of integers {aₙ}, indexed by the natural numbers, such that if n is divisible by m then aₙ is divisible by aₘ.


collineation

collineation (countable and uncountable, plural collineations)

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From Latin collineare to direct in a straight line. See collimation.

(algebra) An isomorphism from one projective space to another, often to itself.


wielomian

pl-noun

wielomian m inan (related adjective wielomianowy)

wielomianowy (adjective)
wielomian (nominative singular)
wielomiany (nominative plural)
wielomianu (genitive singular)
wielomianów (genitive plural)
wielomianowi (dative singular)
wielomianom (dative plural)
wielomian (accusative singular)
wielomiany (accusative plural)
wielomianem (instrumental singular)
wielomianami (instrumental plural)
wielomianie (locative singular)
wielomianach (locative plural)
wielomianie (singular vocative)
wielomiany (plural vocative)

From wiele + -o- + miano. Calque of Latin polynomium.

(algebra) polynomial (expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as a#95;nxⁿ#43;a#95;#123;n-1#125;x#123;n-1#125;#43;...#43;a#95;0x⁰)


лупа

ru-noun+

лу́па • (lúpa) f inan (genitive лу́пы, nominative plural лу́пы, genitive plural луп)

лу́па (canonical)
lúpa (romanization)
лу́пы (genitive)
лу́пы (nominative plural)
луп (genitive plural)
лу́па (nominative singular)
лу́пы (nominative plural)
лу́пы (genitive singular)
лу́п (genitive plural)
лу́пе (dative singular)
лу́пам (dative plural)
лу́пу (accusative singular)
лу́пы (accusative plural)
лу́пой (instrumental singular)
лу́пою (instrumental singular)
лу́пами (instrumental plural)
лу́пе (prepositional singular)
лу́пах (plural prepositional)


connex

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(algebra) A correspondence between a projective space and its dual.


ultraradical

ultraradicals (plural)
ultra-radical (alternative)

LL-Q1860 (eng)-Flame, not lame-ultraradical.wav

(algebra) A root of the polynomial x⁵ + x + a, where a is a complex number.


singulär

de-adj

singulär (strong nominative masculine singular singulärer, not comparable)

singulärer (masculine nominative singular strong)
singulär (masculine predicative singular)
singulär (feminine predicative singular)
singulär (neuter predicative singular)
singulär (plural predicative)
singulärer (masculine nominative singular strong without-article)
singuläre (feminine nominative singular strong without-article)
singuläres (neuter nominative singular strong without-article)
singuläre (nominative plural strong without-article)
singulären (genitive masculine singular strong without-article)
singulärer (feminine genitive singular strong without-article)
singulären (genitive neuter singular strong without-article)
singulärer (genitive plural strong without-article)
singulärem (dative masculine singular strong without-article)
singulärer (dative feminine singular strong without-article)
singulärem (dative neuter singular strong without-article)
singulären (dative plural strong without-article)
singulären (accusative masculine singular strong without-article)
singuläre (accusative feminine singular strong without-article)
singuläres (accusative neuter singular strong without-article)
singuläre (accusative plural strong without-article)
der singuläre (definite includes-article masculine nominative singular weak)
die singuläre (definite feminine includes-article nominative singular weak)
das singuläre (definite includes-article neuter nominative singular weak)
die singulären (definite includes-article nominative plural weak)
des singulären (definite genitive includes-article masculine singular weak)
der singulären (definite feminine genitive includes-article singular weak)
des singulären (definite genitive includes-article neuter singular weak)
der singulären (definite genitive includes-article plural weak)
dem singulären (dative definite includes-article masculine singular weak)
der singulären (dative definite feminine includes-article singular weak)
dem singulären (dative definite includes-article neuter singular weak)
den singulären (dative definite includes-article plural weak)
den singulären (accusative definite includes-article masculine singular weak)
die singuläre (accusative definite feminine includes-article singular weak)
das singuläre (accusative definite includes-article neuter singular weak)
die singulären (accusative definite includes-article plural weak)
ein singulärer (includes-article indefinite masculine mixed nominative singular)
eine singuläre (feminine includes-article indefinite mixed nominative singular)
ein singuläres (includes-article indefinite mixed neuter nominative singular)
singulären (indefinite mixed nominative plural)
keine singulären (includes-article indefinite mixed negative nominative plural)
eines singulären (genitive includes-article indefinite masculine mixed singular)
einer singulären (feminine genitive includes-article indefinite mixed singular)
eines singulären (genitive includes-article indefinite mixed neuter singular)
singulären (genitive indefinite mixed plural)
keiner singulären (genitive includes-article indefinite mixed negative plural)
einem singulären (dative includes-article indefinite masculine mixed singular)
einer singulären (dative feminine includes-article indefinite mixed singular)
einem singulären (dative includes-article indefinite mixed neuter singular)
singulären (dative indefinite mixed plural)
keinen singulären (dative includes-article indefinite mixed negative plural)
einen singulären (accusative includes-article indefinite masculine mixed singular)
eine singuläre (accusative feminine includes-article indefinite mixed singular)
ein singuläres (accusative includes-article indefinite mixed neuter singular)
singulären (accusative indefinite mixed plural)
keine singulären (accusative includes-article indefinite mixed negative plural)

/zɪŋɡuˈlɛːɐ̯/

/zɪŋɡuˈleːɐ̯/

(algebra) singular (of a matrix; having no inverse)


sabit

head

sabit (definite accusative sabiti, plural sabitler)

sabiti (accusative definite)
sabitler (plural)

From Ottoman Turkish ثابت (sabit), from Arabic ثَابِت (ṯābit).


antidivision

en-noun

antidivision (usually uncountable, plural antidivisions)

LL-Q1860 (eng)-Flame, not lame-antidivision.wav

2008, Patrick Bosc, Olivier Pivert, A Family of Tolerant Antidivision Operators for Database Fuzzy Querying Book Series:

It can be noticed that an antidivision is nothing but an antijoin followed by a projection over X […]

2008, Sergio Greco, Thomas Lukasiewicz, Scalable uncertainty management:

(algebra) A particular extension of the antijoin operator.


antijoin

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(algebra, databases) A type of join, denoted by ▷, such that R▷S yields only those tuples in R for which S has no corresponding tuple of the same value.


grúpa

is-noun

grúpa f (genitive singular grúpu, nominative plural grúpur)

grúpu (genitive singular)
grúpur (nominative plural)
grúpa (indefinite nominative singular)
grúpan (definite nominative singular)
grúpur (indefinite nominative plural)
grúpurnar (definite nominative plural)
grúpu (accusative indefinite singular)
grúpuna (accusative definite singular)
grúpur (accusative indefinite plural)
grúpurnar (accusative definite plural)
grúpu (dative indefinite singular)
grúpunni (dative definite singular)
grúpum (dative indefinite plural)
grúpunum (dative definite plural)
grúpu (genitive indefinite singular)
grúpunnar (definite genitive singular)
grúpna (genitive indefinite plural)
grúpnanna (definite genitive plural)


algebră

Borrowed from French algèbre, from Latin algebra.


ordered field

(algebra) A field which has an order relation satisfying these properties: trichotomy, transitivity, preservation of an inequality when the same element is added to both sides, and preservation of an inequality when the same strictly positive element is multiplied to both sides.

corpo ordenado

field satisfying the properties of trichotomy, transitivity and preservation of an inequality when the same element is added to both sides


idealizer

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(algebra) For a ring R and an ideal m of R, the set of all u ∈ Frac(R) such that um ⊂ m, where Frac(R) is the fraction field of R.


kisi

inh

<i class="Latn mention" lang="ms">kisi</i>

(mathematics, algebra, statistics) In any of several technical senses, a substructure with properties analogous to those the set of points with integer coordinates bears in relation to the full Euclidian space these points are contained in

(mathematics) graph that can be drawn within a repeating arrangement of points


scalar multiplication

en-noun

scalar multiplication (countable and uncountable, plural scalar multiplications)

(abstract algebra) multiplication of a module element by a ring element


commutatif

fr-adj

commutatif (feminine commutative, masculine plural commutatifs, feminine plural commutatives)

commutative (feminine)
commutatifs (masculine plural)
commutatives (feminine plural)

(algebra, logic) commutative (such that the order in which the operands are taken does not affect their image under the operation)


nucli

Borrowed from Latin nū̆cleus, diminutive of nux (“nut”).


compositum

LL-Q1860 (eng)-Flame, not lame-compositum.wav

1998, Iain T. Adamson (translator), David Hilbert, The Theory of Algebraic Number Fields, [1897, D. Hilbert, Zahlbericht], Springer, page 98,
Of particular interest is the case in which the discriminants of the fields forming the compositum are relatively prime.

For #92;Q, we get the maximal abelian extension by adjoining all roots of unity, i.e., taking compositum of all m-th cyclotomic fields (this is the famous Kronecker-Weber theorem).

2004, Dinesh S. Thakur, Function Field Arithmetic, World Scientific, page 81:

This is then just the field compositum of all the quadratic extensions F(#92;sqrt#123;a#95;i#125;) in the algebraic closure of F.

2005, T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society, page 333:

(algebra, field theory) Given a field extension F/K and subextensions A and B, the smallest subextension that contains both A and B.

field compositum

smallest subextension of a given field extension that contains two given subextensions


aontéarmach

head

aontéarmach (genitive singular masculine aontéarmaigh, genitive singular feminine aontéarmaí, plural aontéarmacha, not comparable)

aontéarmaigh (genitive masculine singular)
aontéarmaí (feminine genitive singular)
aontéarmacha (plural)
aontéarmach (masculine nominative singular)
aontéarmach (feminine nominative singular)
aontéarmacha (nominative plural)
aontéarmaigh (masculine singular vocative)
aontéarmach (feminine singular vocative)
aontéarmacha (plural vocative)
aontéarmaigh (genitive masculine singular)
aontéarmaí (feminine genitive singular)
aontéarmacha (genitive plural)
aontéarmach (genitive plural)
aontéarmach (dative masculine singular)
aontéarmach (dative feminine singular)
aontéarmaigh (archaic dative feminine singular)
aontéarmacha (dative plural)
níos aontéarmaí (comparative)
is aontéarmaí (superlative)
aontéarmach
n-aontéarmach
haontéarmach

From aon- (“mono-”) + téarma (“term”) + -ach (adjectival suffix).


aonphléacs

aonphléacs m (genitive singular aonphléacs, nominative plural aonphléacsanna)

aonphléacs (genitive singular)
aonphléacsanna (nominative plural)
aonphléacs (indefinite nominative singular)
aonphléacsanna (indefinite nominative plural)
a aonphléacs (indefinite singular vocative)
a aonphléacsanna (indefinite plural vocative)
aonphléacs (genitive indefinite singular)
aonphléacsanna (genitive indefinite plural)
aonphléacs (dative indefinite singular)
aonphléacsanna (dative indefinite plural)
an t-aonphléacs (definite nominative singular)
na haonphléacsanna (definite nominative plural)
an aonphléacs (definite genitive singular)
na n-aonphléacsanna (definite genitive plural)
leis an aonphléacs (dative definite singular)
don aonphléacs (dative definite singular)
leis na haonphléacsanna (dative definite plural)
aonphléacs
n-aonphléacs
haonphléacs
t-aonphléacs

From aon- (“one, uni-, mono-”) + -pléacs (“-plex”).


Boolean ring

Named after George Boole (1815–1864), an English mathematician, educator, philosopher and logician.

From the defining idempotency property of a Boolean ring it is possible to prove that such ring has to have the further properties that each element is its own inverse and that such ring must be commutative.

multiplicative operation
multiplicative operation

(algebra) A ring whose multiplicative operation is idempotent.


multibracket

LL-Q1860 (eng)-Flame, not lame-multibracket.wav

(algebra) A partition of a set into disjoint subsets such that each subset contains an even number of elements, and that if those elements are ordered, they divide the subset into ranges (1st to 2nd element, 3rd to 4th element, and so on) such that no element from another subset appears in any of the ranges.


commutative algebra

en-noun

commutative algebra (countable and uncountable, plural commutative algebras)

noncommutative algebra

concerned with rings that are not assumed to be commutative

Theorem. If the complex of the linearly independent numbers of the form #92;zetaa-a#92;zeta be deleted from an algebra, the remaining numbers form a commutative algebra.

1907, James Byrnie Shaw, Synopsis of Linear Associative Algebra, Carnegie Institution of Washington, page 46:

The existence of commutative algebras of Toeplitz operators is not usual, for example there are not non trivial commutative algebras of Toeplitz operators acting on the Hardy space.

2010, M. Loaiza, A. Sánchez-Numgaray, “On C*-Algebras of Super Toeplitz Operators with Radial Symbols”, in Roland V. Duduchava, Israel Gohberg, Sergei M. Grudsky, Vladimir Rabinovich, editors, Recent Trends in Toeplitz and Pseudodifferential Operators, Springer, page 175:

We devote this chapter to the study of unitary commutative algebras in dg-modules, in simplicial modules and in cosimplicial modules.

2017, Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmuller Groups: Part 2, American Mathematical Society, page 163:

(algebra) Any algebra (mathematical structure) in which the multiplication operation is commutative.

algebra commutativa

algebra in which multiplication is commutative

Abelian algebra

algebra which has a commutative multiplication operation


Heyting algebra

Heyting algebra (plural Heyting algebras)

After Dutch mathematician Arend Heyting, who developed the theory as a way of modelling his intuitionistic logic.

The laws of Heyting algebra embody a rich and profound mathematical structure that is manifest in a variety of contexts. It arises from the epistemological deliberations of Brouwer, the topologisation (localisation) of set-theoretic notions, and the categorial formulation of set theory, all of which, although interrelated, are independently motivated. The ubiquity lends weight, not to the suggestion that the correct logic is in fact intuitionistic instead of classical, but rather to the recognition that thinking in such terms is simply inappropriate — in the same way that it is inappropriate to speak without qualification about the correct geometry.

1984, Robert Goldblatt, Topoi, the categorial analysis of logic, page xii:

which is just obvious.

1994, Francis Borceux, Handbook of Categorical Algebra 3: Categories of Sheaves, Cambridge University Press, page 13, Proposition 1.2.14 should certainly be completed by the observation that the modus ponens holds as well in every Heyting algebra. Since, in the intuitionistic propositional calculus, being a true formula is being a terminal object (see proof of 1.1.3), the modus ponens of a Heyting algebra reduces to a=1 and a<b imply b=1

1997, J. G. Stell, M. W. Worboys, The Algebraic Structure of Sets and Regions, Stephen C. Hirtle, Andrew U. Frank (editors), Spatial Information Theory A Theoretical Basis for GIS: International Conference, Proceedings, Springer, LNCS 1329, page 163,
The main contention of this paper is that Heyting algebras, and related structures, provide elegant and natural theories of parthood and boundary which have close connections to the above three ontologies.

(algebra, order theory) A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called "implies", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b).

Arend Heyting
Heyting algebra
intuitionistic logic

algebra di Heyting

bounded lattice equipped with operation called implies


常數

/ʈ͡ʂʰɑŋ³⁵ ʂu⁵¹/

/sœːŋ²¹ sou̯³³/

/sĩõ⁵⁵⁻¹¹ siau²¹³/

/ʈ͡ʂʰɑŋ³⁵ ʂu⁵¹/

/sœːŋ²¹ sou̯³³/

/sĩõ⁵⁵⁻¹¹ siau²¹³/


mpanasaraka

Agent noun of manasaraka (“decompose; break down”)


orthotomic

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(algebra) imaginary; involving the square root of a negative number.


homamorfacht

head

homamorfacht f (genitive singular homamorfachta)

homamorfachta (genitive singular)
homamorfacht (indefinite nominative singular)
a homamorfacht (indefinite singular vocative)
homamorfachta (genitive indefinite singular)
homamorfacht (dative indefinite singular)
an homamorfacht (definite nominative singular)
na homamorfachta (definite genitive singular)
leis an homamorfacht (dative definite singular)
don homamorfacht (dative definite singular)


iseamorfacht

iseamorfacht f (genitive singular iseamorfachta)

iseamorfachta (genitive singular)
iseamorfacht (indefinite nominative singular)
a iseamorfacht (indefinite singular vocative)
iseamorfachta (genitive indefinite singular)
iseamorfacht (dative indefinite singular)
an iseamorfacht (definite nominative singular)
na hiseamorfachta (definite genitive singular)
leis an iseamorfacht (dative definite singular)
don iseamorfacht (dative definite singular)
iseamorfacht
n-iseamorfacht
hiseamorfacht


osculant

LL-Q1860 (eng)-Flame, not lame-osculant.wav

From Latin osculans, osculantis, present participle of osculari (“to kiss”). See osculate.

(algebra) The condition that the solution to a set of simultaneous quantics, is also the solution of the corresponding set of tangential quantics.


p-adic norm

Let K be a field with discrete valuation v, and #92;mathcal#123;O#125;#95;K be the valuation ring.
Definition 2.3.1 A p'''-adic norm on vector space V over K is a function #92;alpha#58;V#92;rightarrow#92;R satisfying:
a) #92;alpha(x)#92;ge 0 and #92;alpha(x)#61;0 if and only if x#61;0.
b) #92;alpha(ax)#61;#92;verta#92;vert#92;alpha(x) for a#92;inK and x#92;inV.
c) #92;alpha(x#43;y)#92;le#92;operatorname#123;sup#125;(#92;alpha(x),#92;alpha(y)) for x,y#92;inV.
If #92;alpha is a p'''-adic norm and tgt;0, then the dilation t#92;alpha is a p'''-adic norm, and we denote by N#95;p(V) the set of dilation classes of p'''-adic norms on V.

2006, Kang Zuo, Representations of Fundamental Groups of Algebraic Varieties, Springer, page 20:

(algebra) A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of p-adic absolute value).


nomial

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sidoklass

sidoklass (indefinite nominative singular)
sidoklass (genitive indefinite singular)
sidoklassen (definite nominative singular)
sidoklassens (definite genitive singular)
sidoklasser (indefinite nominative plural)
sidoklassers (genitive indefinite plural)
sidoklasserna (definite nominative plural)
sidoklassernas (definite genitive plural)


högersidoklass

högersidoklass (indefinite nominative singular)
högersidoklass (genitive indefinite singular)
högersidoklassen (definite nominative singular)
högersidoklassens (definite genitive singular)
högersidoklasser (indefinite nominative plural)
högersidoklassers (genitive indefinite plural)
högersidoklasserna (definite nominative plural)
högersidoklassernas (definite genitive plural)


vänstersidoklass

vänstersidoklass (indefinite nominative singular)
vänstersidoklass (genitive indefinite singular)
vänstersidoklassen (definite nominative singular)
vänstersidoklassens (definite genitive singular)
vänstersidoklasser (indefinite nominative plural)
vänstersidoklassers (genitive indefinite plural)
vänstersidoklasserna (definite nominative plural)
vänstersidoklassernas (definite genitive plural)


खण्ड

khaṇḍa (romanization)
खण्डः (nominative singular)
खण्डौ (dual nominative)
खण्डा1 (dual nominative)
खण्डाः (nominative plural)
खण्डासः1 (nominative plural)
खण्डम् (accusative singular)
खण्डौ (accusative dual)
खण्डा1 (accusative dual)
खण्डान् (accusative plural)
खण्डेन (instrumental singular)
खण्डाभ्याम् (dual instrumental)
खण्डैः (instrumental plural)
खण्डेभिः1 (instrumental plural)
खण्डाय (dative singular)
खण्डाभ्याम् (dative dual)
खण्डेभ्यः (dative plural)
खण्डात् (ablative singular)
खण्डाभ्याम् (ablative dual)
खण्डेभ्यः (ablative plural)
खण्डस्य (genitive singular)
खण्डयोः (dual genitive)
खण्डानाम् (genitive plural)
खण्डे (locative singular)
खण्डयोः (dual locative)
खण्डेषु (locative plural)
खण्ड (singular vocative)
खण्डौ (dual vocative)
खण्डा1 (dual vocative)
खण्डाः (plural vocative)
खण्डासः1 (plural vocative)
खण्डम् (nominative singular)
खण्डे (dual nominative)
खण्डानि (nominative plural)
खण्डा1 (nominative plural)
खण्डम् (accusative singular)
खण्डे (accusative dual)
खण्डानि (accusative plural)
खण्डा1 (accusative plural)
खण्डेन (instrumental singular)
खण्डाभ्याम् (dual instrumental)
खण्डैः (instrumental plural)
खण्डेभिः1 (instrumental plural)
खण्डाय (dative singular)
खण्डाभ्याम् (dative dual)
खण्डेभ्यः (dative plural)
खण्डात् (ablative singular)
खण्डाभ्याम् (ablative dual)
खण्डेभ्यः (ablative plural)
खण्डस्य (genitive singular)
खण्डयोः (dual genitive)
खण्डानाम् (genitive plural)
खण्डे (locative singular)
खण्डयोः (dual locative)
खण्डेषु (locative plural)
खण्ड (singular vocative)
खण्डे (dual vocative)
खण्डानि (plural vocative)
खण्डा1 (plural vocative)
काण्ड (alternative)
काण्ड (alternative)

An internally-derived word, likely of non-Indo-European, probably Dravidian origin; compare Tamil கண்டு (kaṇṭu, “hard candy”), கண்டம் (kaṇṭam, “piece, part”), Tamil கட்டு (kaṭṭu, “to harden, condense”), Tamil கண்டி (kaṇṭi, “to cut”). Watkins instead suggests a borrowing from Munda, while Mayrhofer mentions a potential connection to क्षद् (kṣad, “to cut, carve”).


展開

/ʈ͡ʂän²¹⁴⁻²¹ kʰaɪ̯⁵⁵/

/t͡siːn³⁵ hɔːi̯⁵⁵/

/tiɛn⁵⁵⁴⁻²⁴ kʰai³³/

/tiɛn⁵³⁻⁴⁴ kʰai⁴⁴/

/tiɛn⁴¹⁻⁴⁴ kʰui⁴⁴/

/tiɛn⁵³⁻⁴⁴ kʰui⁴⁴/

/tiɛn⁵⁵⁴⁻²⁴ kʰui³³/

/tʰiɛn⁵³⁻⁴⁴ kʰui⁴⁴/

/tʰiɛn⁴¹⁻⁴⁴ kʰui⁴⁴/

/tiaŋ⁵²⁻³⁵ kʰui³³/

/ʈ͡ʂän²¹⁴⁻²¹ kʰaɪ̯⁵⁵/

/t͡siːn³⁵ hɔːi̯⁵⁵/

/tiɛn⁵⁵⁴⁻²⁴ kʰai³³/

/tiɛn⁵³⁻⁴⁴ kʰai⁴⁴/

/tiɛn⁴¹⁻⁴⁴ kʰui⁴⁴/

/tiɛn⁵³⁻⁴⁴ kʰui⁴⁴/

/tiɛn⁵⁵⁴⁻²⁴ kʰui³³/

/tʰiɛn⁵³⁻⁴⁴ kʰui⁴⁴/

/tʰiɛn⁴¹⁻⁴⁴ kʰui⁴⁴/

/tiaŋ⁵²⁻³⁵ kʰui³³/

(algebra) to expand (to rewrite (an expression) as a longer, yet equivalent sum of terms)


homomorphous

en-adj

homomorphous (comparative more homomorphous, superlative most homomorphous)

more homomorphous (comparative)
most homomorphous (superlative)

LL-Q1860 (eng)-Flame, not lame-homomorphous.wav

From Ancient Greek ὁμός (homós, “same”) + μορφή (morphḗ, “shape”) + -ous.

(algebra, biology) Characterized by homomorphism.


inverse system

contravariant functor
contravariant functor

(algebra) A set of algebraic structures (which are part of a concrete category; e.g., groups) and a set of morphisms between them (e.g., group homomorphisms) which all together form a small category which is the image of a contravariant functor whose domain is a directed poset.


inverse limit

An inverse limit has “natural projections” which are restrictions of the projections of the Cartesian product (to a domain which is the inverse limit). The reason why the projections are described as “natural” would be the following: besides the functor from an index poset to the inverse system, there is another functor from the same index poset to the inverse limit of that system, this functor being a constant functor. Then there is a natural transformation from the constant functor to the inverse limit’s functor: the components of such natural transformation are the said “natural projections”.

Inverse limits are concrete-categorical versions of limits.

(algebra) A subset of the Cartesian product of all the members of an inverse system, such that a member M of the subset is a sort of “cross section” of the inverse system (as fiber bundle) induced by the morphisms of it. (If i<j in the indexing poset then f_ij:A_j→A_i in the inverse system and if a_i∈A_i, a_j∈A_j are components of M then f_ij(a_j)=a_i).


extension field

en-noun

extension field (plural extension fields)

Suppose F is a subfield of the field K. Then K is called an extension field of F. So, for instance, GF(4) and GF(8) are extension fields of GF(2), although GF(8) is not an extension field of GF(4).

1992, James G. Oxley, “Matroid Theory”, in Paperback, Oxford University Press, published 2006, page 215:

This extension field of F always contains a root of f in the sense that if K#61;F#91;x#93;#47;(f(x)) then x is a root of f(y) in K#91;y#93;. It then follows that any polynomial will have roots, either in the original field of its coefficients or in some extension field.

1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56:

An extension field, by which we mean a bigger field containing F, is automatically a vector space over F. We call it a finite extension if it is a finite vector space. By the degree of a finite extension we mean its dimension as a vector space. One common way of obtaining extension fields is to adjoin an element to F: we say that K#61;F(#92;alpha) if K is the field consisting of all rational expressions formed using #92;alpha and elements of F.

1998, Neal Koblitz, Algebraic Aspects of Cryptography, Volume 3, Springer, page 53:

(algebra, field theory) A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.


Noetherian ring

Noetherian ring (plural Noetherian rings)

Noetherian rings (plural)
noetherian ring (alternative)

/nəˈtɛ.ɹi.ən ˈɹɪŋɡ/

Named after German mathematician Emmy Noether (1882–1935).

The central position occupied by Noetherian rings in commutative ring theory became evident from her^([Noether's]) work.

1986, Hideyuki Matsumura, translated by M. Reid, Commutative Ring Theory, Paperback edition, Cambridge University Press, published 1989, page ix:

In this chapter the focus moves from semiprime rings to general Noetherian rings, although it does concentrate on prime and semiprime ideals.

2000, John C. McConnell, James Christopher Robson, Lance W. Small, Noncommutative Noetherian Rings, 2nd edition, American Mathematical Society, page 97:

2004, K. R. Goodearl, Introduction to the Second Edition, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2nd Edition, page viii,
During this same period, the explosive growth of the area of quantum groups provided a large new crop of noetherian rings to be analyzed, and thus gave major impetus to research in noetherian ring theory.

(algebra, ring theory) A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both left-Noetherian (every left ideal is finitely generated) and right-Noetherian (every right ideal is finitely generated).


algebraic closure

algebraic closure (plural algebraic closures)

The calculation of these various Galois groups leads to a determination of the algebraic closures of the ground fields in the splitting fields of the corresponding vectorial polynomials.

1999, Shreeram S. Abhyankar, “Galois Theory of Semilinear Transformations”, in Helmut Voelklein, David Harbater, J. G. Thompson, Peter Müller, editors, Aspects of Galois Theory, Cambridge University Press, page 1:

It turns out that the algebraic closure #92;mathbf#123;Q#125;#92;mathrm#123;a#125;#95;p is not complete, so we shall consider its completion #92;mathbf#123;C#125;#95;p: This field turns out to be algebraically closed and is a natural domain for the study of "analytic functions."

2000, Alain M. Robert, A Course in p-adic Analysis, Springer, page 127:

While #92;Complex contains an algebraic closure of #92;mathbb#123;Q#125;, it is by no means the only algebraically closed field containing an algebraic closure of #92;mathbb#123;Q#125;. We denote by #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; the algebraic closure of #92;mathbb#123;Q#125; in #92;Complex; this field is simply the subfield of #92;Complex consisting of algebraic numbers. The field #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; is isomorphic, then, to any algebraic closure of #92;mathbb#123;Q#125;, but even knowing that it is unique up to isomorphism very likely leaves us no more familiar with #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; than we were.

2004, John Swallow, Exploratory Galois Theory, Cambridge University Press, page 179:

(algebra, field theory, of a field F) A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).

algebrallinen sulkeuma

field which is algebraically closed over another one


supercatenoid

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(algebra, geometry) a geometric surface constituting a transition between a cube (or similar surface) and a catenoid.


UFD

unique factorization domain
unique factorization domain#English

(algebra) Initialism of unique factorization domain.


unique factorization ring

unique factorization ring (plural unique factorization rings)

(algebra, ring theory) A ring in which every non-zero, non-unit (i.e., proper) element can be factored into a product of prime elements (not necessarily distinct), such that such factorization is unique up to permutation of the factors and the replacing of primes by their associate primes.


unique factorization domain

unique factorization domain (plural unique factorization domains)

unique factorization ring
unique factorization ring

(algebra, ring theory) A unique factorization ring which is also an integral domain.


principal ideal ring

principal ideal ring (plural principal ideal rings)

(algebra, ring theory) A commutative ring in which every ideal is a principal ideal.

(algebra, ring theory) A noncommutative ring in which every left ideal is a principal ideal and so is every right ideal.


commutative ring

commutative ring (plural commutative rings)

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications.

1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129:

As usual, it is simpler to begin by looking at a more general setting—in this case, commutative rings—before getting involved with polynomial rings. It turns out that the nature of the ideals in a commutative ring is important: for example, we have already seen that gcd's exist in PIDs, while this may not be true in other commutative rings.

2002, Joseph J. Rotman, Advanced Modern Algebra, 2nd edition, American Mathematical Society, page 295:

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#92;inP, it follows that a#92;inP or b#92;inP; equivalently, P is a prime ideal if and only if the factor ring R#47;P is a domain.

2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, page 47:

(algebra, ring theory) A ring whose multiplicative operation is commutative.

anello commutativo

a ring whose multiplicative operation is commutative


Dedekind domain

Dedekind domain (plural Dedekind domains)

Named after German mathematician Richard Dedekind (1831–1916).

It can be proved that a Dedekind domain (as defined above) is equivalent to an integral domain in which every proper fractional ideal is invertible.

In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains.

1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Elsevier (Academic Press), page 201:

As we can see every principal ideal domain is a Dedekind domain.

2007, Leonid Kurdachenko, Javier Otal, Igor Ya. Subbotin, Artinian Modules over Group Rings, Springer (Birkhäuser), page 55:

Let us recall some material about Dedekind domains from Chapters VIII and IX of Basic Algebra. A Dedekind domain is a Noetherian integral domain that is integrally closed and has the property that every nonzero prime ideal is maximal. Any Dedekind domain has unique factorization for its ideals.

2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 266:

(algebra, ring theory) An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations).

anneau de Dedekind

integral domain whose prime ideals factorise uniquely


De Morgan algebra

De Morgan algebra (plural De Morgan algebras)

De Morgan algebras (plural)
de Morgan algebra (alternative)

Named after British mathematician and logician Augustus De Morgan (1806–1871). The notion was introduced by Grigore Moisil.

Finally it is shown that the compact elements in the congruence lattice of a De Morgan algebra form a Boolean sublattice.

1980, H. P. Sankappanavar, “A Characterization of Principal Congruences of De Morgan Algebras and its Applications”, in A. I. Arruda, R. Chuaqui, N. C. A. Da Costa, editors, Mathematical Logic in Latin America: Proceedings of the IV Latin American Symposium on Mathematical Logic, page 341:

By a topological de Morgan algebra we shall mean an abstract algebra (A,#92;land,#92;lor,#92;',l) where (A,#92;land,#92;lor,l) is a de Morgan algebra,

2000, Luo Congwen, Topological De Morgan Algebras and Kleene-Stone Algebras: The Journal of Fuzzy Mathematics, Volume 8, Pages 1-524, page 268:

If (L,#92;le,#123;⁻#125;) is a bounded distributive lattice with negation function (resp. a De Morgan algebra), then (L#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle,#92;le,#123;⁻#125;) constitutes also a bounded distributive lattice with negation function (resp. a De Morgan algebra); for every r#92;inL#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle its negation #92;overline#123;r#125;#92;inL#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle is defined by (#92;overline#123;r#125;,s)#61;#92;overline#123;(r,s)#125; for every s#92;inS.

2009, George Rahonis, “Chapter 12: Fuzzy Languages”, in Manfred Droste, Werner Kuich, Heiko Vogler, editors, Handbook of Weighted Automata, Springer, page 486:

distributive lattice
distributive lattice

(algebra, order theory) A bounded distributive lattice equipped with an involution (typically denoted ¬ or ~) which satisfies De Morgan's laws.

Augustus De Morgan
De Morgan algebra
Grigore Moisil


Kleene algebra

(algebra) A De Morgan algebra which also satisfies the inequation x∧∼x⩽y,∨∼y for all x and y, where "∼" here denotes the De Morgan involution.


ordered integral domain

ordered integral domain (plural ordered integral domains)

(algebra) An integral domain which has a subset whose elements are said to be "positive", such that this subset is closed under addition, closed under multiplication, and all elements of the integral domain satisfy a law of trichotomy; namely, that either that element is in the said subset, or it is the zero (additive identity), or its product with −1 (the additive inverse of the multiplicative identity) belongs to the said subset.


incógnita

(algebra) unknown (variable whose value is to be found)


semijoin

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(algebra, databases) A type of join in relational algebra, denoted by ⋉ or ⋊.


maxterm

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All the maxterms in a product of maxterms should have the same variables, although each maxterm should differ from every other one by the pattern of complementation of those variables.

(algebra) A sum that is a logical OR of a set of variables where each individual variable only appears once in the sum, either in complemented or uncomplemented form, so that the value of the sum becomes 0.


Euclidean domain

Euclidean domain (plural Euclidean domains)

So called because for such domains it is possible to apply the Euclidean algorithm in order to obtain greatest common divisors.

(algebra) an integral domain in which division with remainder is possible


permutation group

permutation group (plural permutation groups)

In this chapter we shall be concerned with the relationship between permutation groups and graphs. We begin by explaining how a transitive permutation group may be represented graphically, and then we reverse the process, showing that a graph gives rise to a permutation group.

1979, Norman L. Biggs, A. T. White, Permutation Groups and Combinatorial Structures, page 80:

The Galois group G(L_f /C(x)) is called the monodromy group of f, denoted Mon(f), and viewed as a permutation group on the conjugates of y over C(x).

1996, Helmut Volklein, Groups as Galois Groups: An Introduction, page 47:

Now, groups are axiomatically defined, and the above concept is a permutation group, that is, a subgroup of the symmetric group.[…]The study of finite permutation groups is one of the oldest parts of group theory, motivated initially by its connection with solvability of equations.

2002, Peter J. Cameron, “B.5 Permutation Groups”, in Alexander V. Mikhalev, Günter F. Pilz, editors, The Concise Handbook of Algebra, page 86:

(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition.

transformation group

group whose elements are permutations of a set


adfected

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From the Latin adfectus, variant spelling of affectus (“endowed with, possessed of; affected; impaired, weakened”), whence affect and, indirectly, affected.

(algebra, archaic, of an equation) Having different powers of the unknown quantity in its various terms.


iseamorfach

iseamorfach (masculine nominative singular)
iseamorfach (feminine nominative singular)
iseamorfacha (nominative plural)
iseamorfaigh (masculine singular vocative)
iseamorfach (feminine singular vocative)
iseamorfacha (plural vocative)
iseamorfaigh (genitive masculine singular)
iseamorfaí (feminine genitive singular)
iseamorfacha (genitive plural)
iseamorfach (genitive plural)
iseamorfach (dative masculine singular)
iseamorfach (dative feminine singular)
iseamorfaigh (archaic dative feminine singular)
iseamorfacha (dative plural)
níos iseamorfaí (comparative)
is iseamorfaí (superlative)
iseamorfach
n-iseamorfach
hiseamorfach


二項演算子

ja-noun

二(に)項(こう)演(えん)算(ざん)子(し) • (nikōenzanshi)

二項演算子 (canonical)
nikōenzanshi (romanization)


共軛

/kʊŋ⁵¹⁻⁵³ ˀɤ⁵¹/

/kʊŋ⁵¹⁻⁵³ ˀɤ⁵¹/

Based on the calque of Latin coniugāre and inflected forms, from con- (“共”) + iugum (“軛”).

(algebra, of a complex number) complex conjugate


K-theory

From circa 1960. The K stands for German Klasse (“class”).
The theory developed out of algebraic geometry after the 1957 publication of work by German-born French mathematician Alexander Grothendieck.

algebraic K-theory

study of rings generated by vector bundles

K'''-theory as an independent discipline is a fairly new subject, only about 35 years old.

1994, Jonathan Rosenberg, Algebraic K-Theory and Its Applications, Springer, page 1:

K'''-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C^*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K'''-theory (for C^*-algebras) is a pair of functors, called K₀ and K₁, that to each C^*-algebra A associate two Abelian groups K₀(A) and K₁(A).

2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix:

topological K-theory
topological K-theory

(uncountable, algebra, algebraic geometry, algebraic topology) The study of rings R generated by the set of vector bundles over some topological space or scheme;

The study of rings R generated by the set of vector bundles over some topological space or scheme;
(dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.


dwumian

dwumian (nominative singular)
dwumiany (nominative plural)
dwumianu (genitive singular)
dwumianów (genitive plural)
dwumianowi (dative singular)
dwumianom (dative plural)
dwumian (accusative singular)
dwumiany (accusative plural)
dwumianem (instrumental singular)
dwumianami (instrumental plural)
dwumianie (locative singular)
dwumianach (locative plural)
dwumianie (singular vocative)
dwumiany (plural vocative)

From dwu- + miano. Calque of French binôme.


algebraic notation

en-noun

algebraic notation (usually uncountable, plural algebraic notations)

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notação algébrica

a method for recording chess moves using letters and numbers

(algebra) A method for recording domain and range of a function.


seminormal

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(algebra, of a ring) Being a commutative reduced ring in which, whenever x, y satisfy x³=y², there is s with s²=x and s³=y.


dual number

If f(x) is a polynomial or a power series then its derivative can be obtained “automatically” using dual numbers as follows: f'(x)#61;#123;f(x#43;#92;epsilon)-f(x)#92;over#92;epsilon#125; where f(x#43;#92;epsilon) should be expanded using the algebra of dual numbers. The role of the ε is therein similar to that of an infinitesimal.

In the special case of the algebra #92;mathbf#123;D#125; of dual numbers we get the vertical tangent bundle V.

1993, Ivan Kolář, Peter W. Michor, Jan Slovák, Natural Operations in Differential Geometry, Springer, page 336:

These numbers are now called dual numbers (a term due to Study) and can be defined as expressions of the form a#43;b#92;epsilon, #92;epsilon²#61;0. When this is done as Kotel'nikov and Study showed, the dual distance between two points of this sphere representing two lines, forming an angle #92;varphi and having shortest distance d, is equal to the dual number #92;varphi#43;#92;epsilond, while the motions of Euclidean space are represented by rotations of the dual sphere, and consequently depend on three dual parameters.

1996, Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich, translated by Roger Cooke, Mathematics of the 19th Century: Geometry, Analytic Function Theory, Birkhäuser, page 87:

Thus we investigate natural T-functions defined on T#42;T#123;#92;mathbf#123;D#125;#92;otimesA#125;M to determine all natural operators T#92;toTT#42;Tᴬ for #92;mathbf#123;D#125; denoting the algebra of dual numbers.

2002, Jiří Tomáš, “Some Classification Problems on Natural Bundles Related to Weil Bundles”, in Olga Gil-Medrano, Vicente Miquel, editors, Differential Geometry, Valencia 2001: Proceedings of the International Conference, World Scientific, page 297:

(algebra) An element of an algebra (the algebra of dual numbers) which includes the real numbers and an element ε which satisfies ε ≠ 0 and ε² = 0.


matryca

matrycowy (adjective)
matryca (nominative singular)
matryce (nominative plural)
matrycy (genitive singular)
matryc (genitive plural)
matrycy (dative singular)
matrycom (dative plural)
matrycę (accusative singular)
matryce (accusative plural)
matrycą (instrumental singular)
matrycami (instrumental plural)
matrycy (locative singular)
matrycach (locative plural)
matryco (singular vocative)
matryce (plural vocative)

Borrowed from German Matrize. Doublet of matrix.

(algebra) logical matrix (matrix with entries from the Boolean domain)


орон

mn-noun

орон • (oron) (definite plural орнууд); (regular declension)

oron (romanization)
орнууд (definite plural)
орон (attributive)
орон (indefinite nominative singular)
орнууд (definite nominative plural)
орны (genitive indefinite singular)
орнуудын (definite genitive plural)
орныг (accusative indefinite singular)
орнуудыг (accusative definite plural)
оронд (dative indefinite locative singular)
орнуудад (dative definite locative plural)
орноос (ablative indefinite singular)
орнуудаас (ablative definite plural)
орноор (indefinite instrumental singular)
орнуудаар (definite instrumental plural)
оронтой (comitative indefinite singular)
орнуудтай (comitative definite plural)
оронгүй (indefinite privative singular)
орнуудгүй (definite plural privative)
орон руу (directive indefinite singular)
орнууд руу (definite directive plural)
орноо (indefinite nominative singular)
орнуудаа (definite nominative plural)
орныхоо (genitive indefinite singular)
орнуудынхаа (definite genitive plural)
орныгоо (accusative indefinite singular)
орнуудыгаа (accusative definite plural)
орондоо (dative indefinite locative singular)
орнууддаа (dative definite locative plural)
орноосоо (ablative indefinite singular)
орнуудаасаа (ablative definite plural)
орноороо (indefinite instrumental singular)
орнуудаараа (definite instrumental plural)
оронтойгоо (comitative indefinite singular)
орнуудтайгаа (comitative definite plural)
оронгүйгээ (indefinite privative singular)
орнуудгүйгээ (definite plural privative)
орон руугаа (directive indefinite singular)
орнууд руугаа (definite directive plural)
орных (indefinite singular)
орнуудынх (definite plural)
орныхон (indefinite singular)
орнуудынхан (definite plural)

From Proto-Mongolic *oran, a borrowing from Proto-Turkic *orun. Originally the same word as ор (or), only with the unstable n generalized to all cases.
Compare Dongxiang oron, Kazakh орын (oryn).


Eisenstein integer

en-noun

Eisenstein integer (plural Eisenstein integers)

Named after Gotthold Eisenstein (1823–1852), German mathematician.

To divide an Eisenstein integer a#43;b#92;omega by another Eisenstein integer c#43;d#92;omega, notice that (c#43;d#92;omega)(c#43;d)(c#43;d#92;omega²)#61;c³#43;d³; accordingly multiply both denominator and numerator (of the division expressed as a fraction) by (c#43;d)(c#43;d#92;omega²), then simplify.

(algebra) A complex number of the form a+bω, where a and b are integers and ω is defined by the following two rules: (1) ω³=1 and (2) 1+ω+ω²=0; an element of the Euclidean domain ℤ[ω].


⁻¹

multiplicative inverse
multiplicative inverse

(arithmetic, algebra) reciprocal; multiplicative inverse, inverse


biquaternion

(algebra) Any of the numbers w+xi+yj+zk! where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group.


nepoznanica

sh-noun

nȅpoznanica f (Cyrillic spelling не̏познаница)

nȅpoznanica (canonical feminine)
не̏познаница (Cyrillic)
nepoznanica (nominative singular)
nepoznanice (nominative plural)
nepoznanice (genitive singular)
nepoznanica (genitive plural)
nepoznanici (dative singular)
nepoznanicama (dative plural)
nepoznanicu (accusative singular)
nepoznanice (accusative plural)
nepoznanice (singular vocative)
nepoznanico (singular vocative)
nepoznanice (plural vocative)
nepoznanici (locative singular)
nepoznanicama (locative plural)
nepoznanicom (instrumental singular)
nepoznanicama (instrumental plural)


непознаница

не̏познаница

не̏познаница f (Latin spelling nȅpoznanica)

не̏познаница (canonical feminine)
nȅpoznanica (romanization)
непознаница (nominative singular)
непознанице (nominative plural)
непознанице (genitive singular)
непознаница (genitive plural)
непознаници (dative singular)
непознаницама (dative plural)
непознаницу (accusative singular)
непознанице (accusative plural)
непознанице (singular vocative)
непознаницо (singular vocative)
непознанице (plural vocative)
непознаници (locative singular)
непознаницама (locative plural)
непознаницом (instrumental singular)
непознаницама (instrumental plural)

не поз на ни ца

From не̏познат.


anticomutatividade

pt-noun

anticomutatividade f (plural anticomutatividades)

From anti- + comutatividade or anticomutativo + -idade.

(algebra, physics) anticommutativity (math)


anticomutativo

pt-adj

anticomutativo (feminine anticomutativa, masculine plural anticomutativos, feminine plural anticomutativas)

anticomutativa (feminine)
anticomutativos (masculine plural)
anticomutativas (feminine plural)


R-module

(algebra) The monoid action of a ring R on an abelian group.


(algebra, logic) Boolean value corresponding to truth


skew field

skew fields (plural)
skewfield (alternative)

(algebra, dated) A ring in which every nonzero element has a multiplicative inverse; division ring


binómio

binómios (plural)
binômio (alternative Brazil)

(algebra) binomial (polynomial with two terms)


artiniano

artiniano (feminine artiniana, masculine plural artinianos, feminine plural artinianas)

artiniana (feminine)
artinianos (masculine plural)
artinianas (feminine plural)


associatividade

pt-noun

associatividade f (plural associatividades)

(algebra) associativity (condition of being associative)


classe lateral

classe lateral f (plural classes laterais)

(algebra) coset (a copy of a subgroup multiplied by some element from the parent group)


classe lateral à direita

classe lateral à direita f (plural classes laterais à direita)

Literally, “lateral class to the right”.


classe lateral à esquerda

classe lateral à esquerda f (plural classes laterais à esquerda)


comutativo

pt-adj

comutativo (feminine comutativa, masculine plural comutativos, feminine plural comutativas)

comutativa (feminine)
comutativos (masculine plural)
comutativas (feminine plural)

(algebra, of an algebraic structure) commutative (having a commutative operation)


equação algébrica

pt-noun

equação algébrica f (plural equações algébricas)


estrutura algébrica

estrutura algébrica f (plural estruturas algébricas)

(algebra) algebraic structure (sets with operations)


função algébrica

função algébrica f (plural funções algébricas)

(algebra) algebraic function (function that only uses the operations of addition, subtraction, multiplication, division and raising to a rational power)


Newton's method

(algebra, calculus) A method for finding successively better approximations to the roots (or zeroes) of a real-valued function.


identidade aditiva

identidade aditiva f (plural identidades aditivas)

(algebra) additive identity (element which is an identity for an additive operation)


線性方程

ㄒㄧㄢˋ ㄒㄧㄥˋ ㄈㄤ ㄔㄥˊ

ㄒㄧㄢˋ ㄒㄧㄥˋ ㄈㄤ ㄔㄥˊ

сяньсин фанчэн

/ɕi̯ɛn⁵¹⁻⁵³ ɕiŋ⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/siːn³³ sɪŋ³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/ɕi̯ɛn⁵¹⁻⁵³ ɕiŋ⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/siːn³³ sɪŋ³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/


一次方程

/i⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/jɐt̚⁵ t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/i⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/jɐt̚⁵ t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/


三次方程

ㄙㄢ ㄘˋ ㄈㄤ ㄔㄥˊ

ㄙㄢ ㄘˋ ㄈㄤ ㄔㄥˊ

саньцы фанчэн

/sän⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/saːm⁵⁵ t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/sän⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/saːm⁵⁵ t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

(algebra, Mainland China, Hong Kong) cubic equation


Coxeter-Dynkin diagram

en-noun

Coxeter-Dynkin diagram (plural Coxeter-Dynkin diagrams)

After mathematicians H. S. M. Coxeter and Eugene Dynkin.

A Coxeter-Dynkin diagram encodes the information in a Coxeter matrix, which in turn encodes the presentation of a Coxeter group.

Vertices of a Coxeter-Dynkin diagram represent generators of the corresponding Coxeter group. The generators (algebraic) in turn correspond to the reflecting hyperplanes (geometric). A pair of vertices which are not linked by an edge correspond to a pair of commuting generators. An unnumbered edge between a pair of vertices means that the braid relation between the corresponding generators has length three (e.g., aba = bab if the generators are a and b). An edge numbered ≥4 means that the braid relation between the corresponding generators has a length equal to that number. For example, if the edge is numbered 4 then the braid relation is cdcd = dcdc if the generators are c and d. If a set of edges form a cycle then the Coxeter group can be shown to be infinite. If a tree in a Coxeter-Dynkin diagram has more than one numbered edge then the Coxeter group can be shown to be infinite. There are a few more such rules, which ensure that finite Coxeter groups have Coxeter-Dynkin diagrams with relatively simple shapes.

Let 𝒬 be an indecomposable root lattice and let Γ denote the Coxeter-Dynkin diagram of the underlying root system Δ.

1995 June, R. V. Moody, J. Patera, “Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices”, in Canadian Journal of Mathematics, page 597:

There is a deep and only partially understood connection between the classification and structure of singularities and the Coxeter-Dynkin diagrams introduced by H. S .M. Coxeter for classification of reflection-generated groups and by E. B. Dynkin for classification of semisimple Lie algebras.

2000, Andrei Gabrielov, “Coxeter-Dynkin diagrams and singularities”, in Evgeniĭ Borisovich Dynkin, A. A. Yushkevich, Gary M. Seitz, A. L. Onishchik, editors, Selected Papers of E. B. Dynkin with Commentary, page 367:

For 3 ≤ n ≤ 5, we will use Eₙ to denote the Coxeter-Dynkin diagrams of types A₁ + A₂(N = 3), A₄(N = 4) and D₅(N = 5).

2012, Igor V. Dolgachev, Classical Algebraic Geometry: A Modern View, page 363:

(geometry, algebra) A graph with numerically labelled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).

Coxeter-Dynkin diagram
Eugene Dynkin
Harold Scott MacDonald Coxeter


polinomial

pt-adj

polinomial m or f (plural polinomiais, not comparable)

(algebra) polynomial (relating to polynomials)


polinomial

ro-adj

polinomial m or n (feminine singular polinomială, masculine plural polinomiali, feminine and neuter plural polinomiale)

polinomială (feminine singular)
polinomiali (masculine plural)
polinomiale (feminine neuter plural)
polinomial (indefinite masculine neuter singular)
polinomială (feminine indefinite singular)
polinomiali (indefinite masculine plural)
polinomiale (feminine indefinite neuter plural)
polinomialul (definite masculine neuter singular)
polinomiala (definite feminine singular)
polinomialii (definite masculine plural)
polinomialele (definite feminine neuter plural)
polinomiale (feminine indefinite singular)
polinomialului (definite masculine neuter singular)
polinomialei (definite feminine singular)
polinomialilor (definite masculine plural)
polinomialelor (definite feminine neuter plural)


polinomial

es-adj

polinomial m or f (masculine and feminine plural polinomiales)

polinomiales (feminine masculine plural)


polinómio

polinómios (plural)
polinômio (alternative Brazil)

(algebra) polynomial (expression consisting of a sum of terms being the product of a constant coefficient and variable raised to a non-negative integer power)


algoritmo de Euclides

es-noun

algoritmo de Euclides m (plural algoritmos de Euclides)


carrier set

(algebra) the underlying set of an algebraic structure


Monster group

We next consider the possible #92;Z#95;n orbifoldings of #92;mathcal#123;V#125;² with respect to elements of the Monster group M [Tul].

1996, Michael P. Tuite, “Monstrous Moonshine and orbifolds”, in K.T. Arasu, J.F. Dillon, K. Harada, S. Sehgal, R. Solomon, editors, Groups, Difference Sets, and the Monster: Proceedings of a Special Research Quarter, Walter de Gruyter, page 443:

We note that c(1)#61;196883; this is not just a numerological coincidence, but follows from a branch of mathematics called moonshine theory which arises from string theory and relates the Monster group and modular functions.

2008, L. J. P. Kilford, Modular Forms: A Classical and Computational Introduction, Imperial College Press, page 106:

They were also crucial in understanding the Monster and Moonshine phenomena, which refers to the fact that the dimensions of the irreducible representations of the largest sporadic finite group, the monster group, show up in the q-expansion coefficients of the elliptic modular function j.

2014, Martin Schlichenmaier, Krichever-Novikov Type Algebras: Theory and Applications, Walter de Gruyter, page 340:

(algebra) The largest sporadic group, of order 2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 (approximately 8 · 10⁵³), denoted M or F₁.


trinómio

trinómios (plural)
trinômio (alternative Brazil)

(algebra) trinomial (a polynomial consisting of three terms)


二項演算

二項演算 (canonical)
nikōenzan (romanization)


cosingular

LL-Q1860 (eng)-Flame, not lame-cosingular.wav

(algebra) Having the same singularities.


niewiadoma

niewiadoma (nominative singular)
niewiadome (nominative plural)
niewiadomej (genitive singular)
niewiadomych (genitive plural)
niewiadomej (dative singular)
niewiadomym (dative plural)
niewiadomą (accusative singular)
niewiadome (accusative plural)
niewiadomą (instrumental singular)
niewiadomymi (instrumental plural)
niewiadomej (locative singular)
niewiadomych (locative plural)
niewiadoma (singular vocative)
niewiadome (plural vocative)

(nominalized, algebra) unknown (variable (usually x, y, or z) whose value is to be found)


כינס

he-verb

כינס / כִּנֵּס • (kinés) (pi'el construction, passive counterpart כונס / כֻּנַּס)

כִּנֵּס (canonical)
kinés (romanization)
כונס (passive)
כֻּנַּס (passive)
לְכַנֵּס
כינוס (noun-from-verb)
כִּנּוּס (noun-from-verb)
כינסתי (feminine first-person masculine past singular)
כִּנַּסְתִּי (feminine first-person masculine past singular)
כינסנו (feminine first-person masculine past plural)
כִּנַּסְנוּ (feminine first-person masculine past plural)
כינסת (masculine past second-person singular)
כִּנַּסְתָּ (masculine past second-person singular)
כינסת (feminine past second-person singular)
כִּנַּסְתְּ (feminine past second-person singular)
כינסתם (masculine past plural second-person)
כִּנַּסְתֶּם (masculine past plural second-person)
כינסתן (feminine past plural second-person)
כִּנַּסְתֶּן (feminine past plural second-person)
כינס (masculine past singular third-person)
כִּנֵּס (masculine past singular third-person)
כינסה (feminine past singular third-person)
כִּנְּסָה (feminine past singular third-person)
כינסו (feminine masculine past plural third-person)
כִּנְּסוּ (feminine masculine past plural third-person)
מְכַנֵּס (masculine present singular)
מְכַנֶּסֶת (feminine present singular)
מְכַנְּסִים (masculine plural present)
מְכַנְּסוֹת (feminine plural present)
אֲכַנֵּס (feminine first-person future masculine singular)
נְכַנֵּס (feminine first-person future masculine plural)
תְּכַנֵּס (future masculine second-person singular)
תְּכַנְּסִי (feminine future second-person singular)
תְּכַנְּסוּ (future masculine plural second-person)
תְּכַנֵּסְנָה (feminine future plural second-person)
יְכַנֵּס (future masculine singular third-person)
תְּכַנֵּס (feminine future singular third-person)
יְכַנְּסוּ (future masculine plural third-person)
תְּכַנֵּסְנָה (feminine future plural third-person)
כַּנֵּס (imperative masculine singular)
כַּנְּסִי (feminine imperative singular)
כַּנְּסוּ (imperative masculine plural)
כַּנֵּסְנָה (feminine imperative plural)


Galois group

Named after Évariste Galois, who first discovered them.

In this section, we show how to determine the Galois group and the roots of an irreducible polynomial of degree 2, 3, or 4.

1996, Patrick Morandi, Field and Galois Theory, Springer, page 123:

Let B be a Galois algebra over R with Galois group G, C the center of B, and K#61;#92;#123;g#92;inG#92;vertg(c)#61;c#92;#92;mathsf#123;for#92;each#125;#92;c#92;inC#92;#125;. A natural question is whether B is a central Galois algebra with Galois group K.

2004, George Szeto, Liangyong Xue, “On Central Galois Algebras of a Galois Algebra”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, CRC Press, page 493:

Our work actually gives an algorithm for computing Galois groups of polynomials f(X)#92;in#92;Q#91;X#93;:

2009, Steven H. Weintraub, Galois Theory, 2nd edition, Springer, page 128:

(algebra, Galois theory) The automorphism group of a Galois extension.

زمرة غالوا

specific group associated with a field extension


קבוצי

he-adj

קִבּוּצִיוֹת

קיבוצי / קִבּוּצִי • (kibutsí) (feminine קיבוצית / קִבּוּצִית, masculine plural קיבוציים / קִבּוּצִיִם, feminine plural קיבוציות / קִבּוּצִיוֹת)

קיבוצי (canonical)
קִבּוּצִי (canonical)
kibutsí (romanization)
קיבוצית (feminine)
קִבּוּצִית (feminine)
קיבוציים (masculine plural)
קִבּוּצִיִם (masculine plural)
קיבוציות (feminine plural)
קִבּוּצִיוֹת (feminine plural)

קִבּוּץ (kibuts, “kibbutz”) + ־י (-í, “-ic, -ish, -like”, used to form adjectives from nouns, demonyms and language names from place names, and so on)


reticulado completo

pt-noun

reticulado completo m (plural reticulados completos)

(algebra) complete lattice (partially ordered set)


change sides, change signs

(algebra, mnemonic) An acronym to help remember that when a value is moved to the opposite side of the equals sign, then the operator is changed also.


homomorfismo

(algebra) homomorphism (structure-preserving map between two algebraic structures)


synthetic division

en-noun

synthetic division (countable and uncountable, plural synthetic divisions)

(algebra) A method of polynomial long division in which one does not write the variables and some of the calculations.

división sintética

a method of polynomial long division in which one does not write the variables and some of the calculations


Aut

(algebra, category theory) The set of automorphisms of a specific object.


bình phương

[ʔɓïŋ˨˩ fɨəŋ˧˧]

[ʔɓɨn˦˩ fɨəŋ˧˧]

[ʔɓɨn˨˩ fɨəŋ˧˧]

Hiệu các bình phương của hai số bằng tích tổng và hiệu của hai số đó

The difference of the squares of two numbers is the product of their sum and their difference.


bình phương

[ʔɓïŋ˨˩ fɨəŋ˧˧]

[ʔɓɨn˦˩ fɨəŋ˧˧]

[ʔɓɨn˨˩ fɨəŋ˧˧]

Two squared equals four


lập phương

[ləp̚˧˨ʔ fɨəŋ˧˧]

[ləp̚˨˩ʔ fɨəŋ˧˧]

[ləp̚˨˩˨ fɨəŋ˧˧]

khối lập phương

cube, regarding the shape and the volume


lập phương

[ləp̚˧˨ʔ fɨəŋ˧˧]

[ləp̚˨˩ʔ fɨəŋ˧˧]

[ləp̚˨˩˨ fɨəŋ˧˧]

Three cubed equals eight


presemifield

LL-Q1860 (eng)-Flame, not lame-presemifield.wav

multiplicative identity
multiplicative identity#English

(mathematics, algebra) A finite algebra satisfying all the axioms for a skew field except multiplicative associativity and the existence of a multiplicative identity.


Henselian

From Hensel (surname) + -ian; after German mathematician Kurt Hensel (1861–1941).

1989, N. Bourbaki, Commutative Algebra: Chapters 1-7, [1985, N. Bourbaki, Éléments de Mathématique Algébre Commutative 1-4 et 5-7, Masson], Springer, page 256,
A local ring satisfying conditions (H) and (C) is called Henselian. Every complete Hausdorff local ring is Henselian. If A is Henselian and B is a commutative A-algebra which is a local ring and a finitely generated A-module, then B is Henselian.

An analogous result holds for Henselian fields. Recall that a field K is said to be Henselian with respect to a valuation v if v has a unique extension (also denoted v) to every algebraic extension of K.[…]It follows that every algebraic extension of an^([sic]) Henselian field K is Henselian.
[…] Every valued field (K,v) has a minimal separable algebraic extension (K#95;v,v) which is Henselian. The valued field is unique up to a K-isomorphism and is called the Henselian closure of (K,v) [Ribenboim, p. 176].

2008, Michael D. Fried, Moshe Jarden, Field Arithmetic, 3rd edition, Springer, page 203:

Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years.

2017, Arno Fehm, Franziska Jahnke, “Recent progress on definability of Henselian valuations”, in Fabrizio Broglia, Françoise Delon, Max Dickman, Danielle Gondard-Cozette, Victoria Ann Powers, editors, Ordered Algebraic Structures and Related Topics: International Conference, American Mathematical Society, page 136:

(algebra, of a ring or field) Which satisfies the criteria for (some formulation of) Hensel's lemma.


plethysm

LL-Q1860 (eng)-Flame, not lame-plethysm.wav

Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm.

2015, Thomas Kahle, Mateusz Michalek, “Obstructions to combinatorial formulas for plethysm”, in arXiv:

(algebra) A particular group-theoretic operation on a set of functions of a given symmetry type.


HDA

HDA (countable and uncountable, plural HDAs)

(algebra) Initialism of higher-dimensional algebra.


정식

jeongsik (romanization)
整式 (hanja)
온식(式) (alternative North-Korea)
옹근식(式) (alternative North-Korea)
완식(完式) (alternative North-Korea)

[ˈt͡ɕɘ(ː)ŋɕʰik̚]

Sino-Korean word from 整式, from 整 (“orderly, neat”) + 式 (“math formula”).

ko-sino

(algebra) algebraic expression with only addition, subtraction or multiplication operations, without any variable in the denominator or the radical sign


corpo ordenado

pt-noun

corpo ordenado m (plural corpos ordenados)

(algebra) ordered field (field satisfying the properties of trichotomy, transitivity and preservation of an inequality when the same element is added to both sides)


संक्रमण

saṃkrámaṇa (romanization)
संक्रमणम् (nominative singular)
संक्रमणे (dual nominative)
संक्रमणानि (nominative plural)
संक्रमणा1 (nominative plural)
संक्रमणम् (accusative singular)
संक्रमणे (accusative dual)
संक्रमणानि (accusative plural)
संक्रमणा1 (accusative plural)
संक्रमणेन (instrumental singular)
संक्रमणाभ्याम् (dual instrumental)
संक्रमणैः (instrumental plural)
संक्रमणेभिः1 (instrumental plural)
संक्रमणाय (dative singular)
संक्रमणाभ्याम् (dative dual)
संक्रमणेभ्यः (dative plural)
संक्रमणात् (ablative singular)
संक्रमणाभ्याम् (ablative dual)
संक्रमणेभ्यः (ablative plural)
संक्रमणस्य (genitive singular)
संक्रमणयोः (dual genitive)
संक्रमणानाम् (genitive plural)
संक्रमणे (locative singular)
संक्रमणयोः (dual locative)
संक्रमणेषु (locative plural)
संक्रमण (singular vocative)
संक्रमणे (dual vocative)
संक्रमणानि (plural vocative)
संक्रमणा1 (plural vocative)

/sɐŋ.kɾɐ́.mɐ.ɳɐ/

/s̪ɐŋ.kɾɐ.mɐ.ɳɐ/

From सम्- (sam-) + क्रम् (kram) + -अन (-ana).

(algebra, mathematics) concurrence (said to be a general designation of a particular class of problems)


vertex form

en-noun

vertex form (countable and uncountable, plural vertex forms)

Named because of its use in finding the vertex of a parabola: (h,k).

(mathematics, algebra) a quadratic equation in the form y=a(x-h)²+k


стала

uk-noun

ста́ла • (stála) f inan (genitive ста́лої, nominative plural ста́лі, genitive plural ста́лих)

ста́ла (canonical)
stála (romanization)
ста́лої (genitive)
ста́лі (nominative plural)
ста́лих (genitive plural)
ста́ла (nominative singular)
ста́лі (nominative plural)
ста́лої (genitive singular)
ста́лих (genitive plural)
ста́лій (dative singular)
ста́лим (dative plural)
ста́лу (accusative singular)
ста́лі (accusative plural)
ста́лою (instrumental singular)
ста́лими (instrumental plural)
ста́лій (locative singular)
ста́лих (locative plural)
ста́ла (singular vocative)
ста́лі (plural vocative)


तावत्

head

तावत् • (tāvat)

sa-adv

तावत् • (tāvat)

tāvat (romanization)
तावन्त् (alternative)

Pali: tāva

From Proto-Indo-European *teh₂wot. Compare Ancient Greek τῆος (têos) / τέως (téōs, “so long”), Latin tantus (“so much, so great”).


क्षय

kṣayá (romanization)
क्षयः (nominative singular)
क्षयौ (dual nominative)
क्षया1 (dual nominative)
क्षयाः (nominative plural)
क्षयासः1 (nominative plural)
क्षयम् (accusative singular)
क्षयौ (accusative dual)
क्षया1 (accusative dual)
क्षयान् (accusative plural)
क्षयेण (instrumental singular)
क्षयाभ्याम् (dual instrumental)
क्षयैः (instrumental plural)
क्षयेभिः1 (instrumental plural)
क्षयाय (dative singular)
क्षयाभ्याम् (dative dual)
क्षयेभ्यः (dative plural)
क्षयात् (ablative singular)
क्षयाभ्याम् (ablative dual)
क्षयेभ्यः (ablative plural)
क्षयस्य (genitive singular)
क्षययोः (dual genitive)
क्षयाणाम् (genitive plural)
क्षये (locative singular)
क्षययोः (dual locative)
क्षयेषु (locative plural)
क्षय (singular vocative)
क्षयौ (dual vocative)
क्षया1 (dual vocative)
क्षयाः (plural vocative)
क्षयासः1 (plural vocative)

→ Old Marathi: क्षय (kṣaya)

→ Marathi: क्षय (kṣay)

Paisaci Prakrit:

Punjabi: ਖੈ (khai)

Pali: khaya (“decay, ruin”)

→ Thai: ขยะ (kà-yà, “rubbish”)

Prakrit khaya

→ Tamil: கயம் (kayam)

Assamese: ক্ষয় (khoy) / খয় (khoy)

Bengali: ক্ষয় (khoẏ)

Konkani: क्षय (kṣay)

Nepali: क्षय (kṣaya)

Odia: କ୍ଷୟ (kṣaya)

Marwari: खे (khe)

Sindhi: کَيِٴ

Hindi: खै (khai)

→ Telugu: క్షయ (kṣaya)

→ Kannada: ಕ್ಷಯ (kṣaya)

From Proto-Indo-Aryan *gẓʰayás, from Proto-Indo-Iranian *gžʰayás, from Proto-Indo-European *dʰgʷʰey- (“to perish, decline”). Cognate with Ancient Greek φθῐ́σῐς (phthĭ́sĭs, “decline, atrophy”), Latin situs (“decay, filth; neglect”) Old English dwīnan (whence English dwindle).


notație algebrică

ro-noun

notație algebrică f (plural notații algebrice)


Galois field

Named after French mathematician Évariste Galois (1811–1832).

named-after

The Galois field #92;mathrm#123;GF#125;(pⁿ) has order pⁿ and characteristic p.

The Galois field #92;mathrm#123;GF#125;(pⁿ) is a finite extension of the Galois field #92;mathrm#123;GF#125;(p) and the degree of the extension is n.

The multiplicative subgroup of a Galois field is cyclic.

A Galois field #92;mathbb#123;F#125;#95;#123;pⁿ#125; is isomorphic to the quotient of the polynomial ring #92;mathbb#123;F#125;#95;p adjoin x over the ideal generated by a monic irreducible polynomial of degree n. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: #92;mathbb#123;F#125;#95;#123;pⁿ#125;#92;cong#123;#92;mathbb#123;F#125;#95;p#91;x#93;#92;over(#92;hatf#95;n(x))#125;.

1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page,
A field with a finite number of elements is called a Galois field.
The number of elements of the prime field k contained in a Galois field K is finite, and is therefore a natural prime p.

The case of most interest to us will be that in which F is a finite field, the Galois field GF(q) for some prime power q. If q is prime, this field is #92;mathbb#123;Z#125;#95;q, the integers 0,1,#92;dots,q-1 with arithmetic modulo q.

2001, Joseph E. Bonin, A Brief Introduction To Matroid Theory, retrieved 2016-05-05:

2006, Debojyoti Battacharya, Debdeep Mukhopadhyay, D. RoyChowdhury, A Cellular Automata Based Approach for Generation of Large Primitive Polynomial and Its Application to RS-Coded MPSK Modulation, Samira El Yacoubi, Bastien Chopard, Stefania Bandini (editors), Cellular Automata: 7th International Conference, Proceedings, Springer, LNCS 4173, page 204,
Generation of large primitive polynomial over a Galois field has been a topic of intense research over the years. The problem of finding a primitive polynomial over a Galois field of a large degree is computationaly expensive and there is no deterministic algorithm for the same.

(algebra) A finite field; a field that contains a finite number of elements.


symmetry group

[The] situation is more delicate in the case of higher dimensional symmetry groups; it is not in general possible to reduce the order of an equation invariant under an r-parameter symmetry group by r using only quadratures.

1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, published 2000, page 76:

It is often assumed that the design unit inside each fundamental region (particularly for monotranslational symmetry groups p111 and p1a1 and ditranslational symmetry groups p1 and pg) is asymmetric.

2000, C. E. Horne, Geometric Symmetry in Patterns and Tilings, page 79:

The structure of the symmetry group of a molecule affects several spectroscopic aspects and vice versa. Thus, it is clearly important to know the possible symmetries of fullerenes and similar structures.
In this chapter, we study symmetry groups of fulleroids.

2010, Stanislav Jendrol, František Kardoš, “28: Symmetry of Fulleroids”, in Klaus D. Sattler, editor, Handbook of Nanophysics: Clusters and Fullerenes, pages 28–1:

(geometry, algebra, group theory) A group whose elements are all the transformations under which a given object remains invariant and whose group operation is function composition.

full symmetry group

emphasises the inclusion of orientation-reversing transformations


Coxeter diagram

en-noun

Coxeter diagram (plural Coxeter diagrams)

It suffices to make use of the tables above and observe that the Coxeter diagram of a direct sum of C-matrices is the disconnected union of the Coxeter diagrams of the summands.
For the description of C⁻-matrices satisfying condition (2) we shall use Coxeter diagrams with the additional stipulation that in case aᵢⱼ < −1 the vertices vᵢ and vⱼ are joined by a dotted line with index −aᵢⱼ. By the Coxeter diagram of a Coxeter polyhedron and a lattice generated by reflections we mean the Coxeter diagram of the corresponding C⁻-matrix.

1991, Gregori A. Margulis, Discrete Subgroups of Semisimple Lie Groups, page 358:

An automorphism of a Coxeter diagram M leads in a natural way to a Coxeter subgroup of the Coxeter group of type M. We introduce admissible partitions of Coxeter diagrams in order to generalize this situation.

1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. Buekenhout, J. Doyen, F. de Clerck, J. A. Thas, J. W. P. Hirschfeld, editors, Finite Geometries and Combinatorics, 2nd International Conference, page 277:

The Coxeter diagrams of type E₇ and E₈ have no nontrivial automorphisms, so σ₀ is trivial in those cases.[…]The Coxeter diagram of type F₄ has a unique nontrivial automorphism, but σ₀ is trivial in this case. One can see this by using the Dynkin diagram instead of the Coxeter diagram and noting that it does not have any nontrivial automorphisms.

2008, Peter Abramenko, Kenneth S. Brown, Buildings: Theory and Applications, page 259:

Coxeter-Dynkin diagram
Coxeter-Dynkin diagram

(geometry, algebra) A Coxeter-Dynkin diagram.


חד איבר

he-noun

חַד אֵיבָר • (khad eivár) m

חַד אֵיבָר (canonical)
khad eivár (romanization)

Calque of English monomial, חַד־ (khad-, “mono-”) + אֵיבָר (“object”).


semiartinian

LL-Q1860 (eng)-Flame, not lame-semiartinian.wav

(algebra) Of a module _RM: such that, for every M→N epimorphism, where N≠0, the socle of N is essential in N.


inverse semigroup

en-noun

inverse semigroup (plural inverse semigroups)

Examples of regular semigroups include any band, inverse semigroup or completely regular semigroup (see sections A. 2, A. 11, and A. 10); in particular, any group is a regular semigroup.

2002, Peter G. Trotter, “A.9: Regular Semigroups”, in Aleksandr Vasilʹevich Mikhalev, Günter Pilz, editors, The Concise Handbook of Algebra, page 35:

Inverse semigroups form a class of semigroups which are closest (in some sense) to groups.

2008, Olexandr Ganyushkin, Volodymyr Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, page v:

Inverse semigroups are central to modern semigroup theory: arguably, they form the most-studied class of semigroups.

2014, Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, page 249:

(algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy.

inverse Halbgruppe

semigroup in which every element has an inverse


Kantor-Koecher-Tits construction

Introduced by Jacques Tits (1962), Kantor (1964), and Koecher (1967).

(algebra) A method of constructing a Lie algebra from a Jordan algebra.


bicomplex number

bicomplex number (plural bicomplex numbers)

(algebra) A pair (w,z) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^*=(w,-z), and the product of two bicomplex numbers as (u,v)(w,z)=(uw-vz,uz+vw).