free

en-adj

free (comparative freer or more free, superlative freest or most free)

freer (comparative)
more free (comparative)
freest (superlative)
most free (superlative)

From Middle English free, fre, freo, from Old English frēo (“free”), from Proto-West Germanic *frī, from Proto-Germanic *frijaz (“beloved, not in bondage”), from Proto-Indo-European *priHós (“pleased, loved”), from *preyH- (“to please, love”). Related to friend.
cognates, etc
Germanic cognates include West Frisian frij (“free”), Dutch vrij (“free”), Low German free (“free”), German frei (“free”), Danish, Swedish and Norwegian fri (“free”).
Other cognates include Sanskrit प्रिय (priyá-, “beloved”).
Germanic and Celtic are the only Indo-European language branches in which the PIE word with the meaning of "dear, beloved" acquired the additional meaning of "free" in the sense of "not in bondage". This was an extension of the idea of "characteristic of those who are dear and beloved", in other words friends and tribe members (in contrast to unfree inhabitants from other tribes and prisoners of war, many of which were among the slaves – compare the Latin use of liberi to mean both "free persons" and "children of a family").

inh

there ain't no such thing as a free lunch

why buy the cow when you can get the milk for free

en:unconstrained, abstractly
en:albegraically generic or universal

(algebra, of an algebraic structure) In any of various technical senses generic, universal.

(most generally, category theory, of an object F in a concrete category, with respect to a set X and a map i from X to the underlying set of F) Such that any map f from X to the underlying set of an object A in the same category as F induces a map ̄f from F to A which is compatible with f (i.e. such that f=̄f∘i).

The fundamental group of the figure eight is free of rank 2.

(group theory, of a group) Having a set of generators which satisfy no non-trivial relations; equivalently, being the group of reduced words on a set of generators.

linearly independent
linearly independent

(commutative algebra, of a module) Having a linearly independent set of generators (called a basis).


zero

en-noun

zero (countable and uncountable, plural zeros or zeroes)

pole

antonym(s) of “value of a function's variables at zero”

From Spanish zero (now cero), Middle French zero (modern French zéro), and their etymon Italian zero, from Medieval Latin zephirum, from Arabic صِفْر (ṣifr, “nothing, cipher”), itself calqued from (semantic loan) Sanskrit शून्य (śūnya, “void, nothingness”). Doublet of cipher and chiffre.

der

Since a commutative zero is the inverse of any additive identity, it must be unique when it exists.

The zero (of a ring or field) has the property that the product of the zero with any element yields the zero.

The quotient ring over a maximal ideal is a field with a single zero element.

(mathematics, algebra) The additive identity element of a monoid or greater algebraic structure, particularly a group or ring.


transitive

From Latin trānsitīvus, from trānsitus, from trāns (“across”) + itus, from eō (“to go”).

(algebra, of a group action) Such that, for any two elements of the acted-upon set, some group element maps the first to the second.


index

en-noun

index (plural indexes or indices or (obsolete, in use in the 17th century) index's)

indexes (plural)
indices (plural)
index's (obsolete plural)

⇒ Chinese: 引得 (yǐndé)

→ Japanese: インデックス (indekkusu)

From Latin index (“a discoverer, informer, spy; of things, an indicator, the forefinger, a title, superscription”), from indicō (“point out, show”); see indicate.

der

The index of 2ℤ in ℤ is 2.

(algebra, index of a subgroup) The number of cosets that exist.


rational

en-adj

rational (comparative more rational, superlative most rational)

more rational (comparative)
most rational (superlative)
rationall (alternative obsolete)

From Old French rationel, rational, from Latin rationalis (“of or belonging to reason, rational, reasonable; having a ratio”), from ratio (“reason; calculation”).

der

/ˈɹæʃ(ə)nəl/

(mathematics, not comparable) Comprising, or expressible as, a ratio

(algebra) Of an algebraic expression in indeterminates, or more generally a function: capable of being expressed as the ratio of two polynomials.


affirmative

en-adj

affirmative (comparative more affirmative, superlative most affirmative)

more affirmative (comparative)
most affirmative (superlative)

/əˈfɜːmətɪv/

From Middle English affirmative, affirmatyve, from Old French affirmatif, from Latin affirmativus, from affirmare (“to assert”).


complete

complete (comparative more complete or completer, superlative most complete or completest)

more complete (comparative)
completer (comparative)
most complete (superlative)
completest (superlative)
compleat (alternative archaic)

From Middle English compleet (“full, complete”), borrowed from Old French complet or Latin completus, past participle of compleō (“I fill up, I complete”) (whence also complement, compliment), from com- + pleō (“I fill, I fulfill”) (whence also deplete, replete, plenty), ultimately from Proto-Indo-European *pleh₁- (“to fill”) (English full).

пълен

of a proof system: such that any semantically valid formula is also provable

complete androgen insensitivity syndrome

(algebra, of a lattice) In which every set with a lower bound has a greatest lower bound.

kompleta

of a lattice: such that every set with a lower bound has a greatest lower bound


unit

Formerly unite, a later form of unity; see unity.

An element 𝑒 of 𝑆 such that 𝑒𝑥 = 𝑥 = 𝑥𝑒 for all 𝑥 ∈ 𝑆 is called a unit element. (When the law of composition is written additively, the unit element is denoted by 0, and is called a zero element.)

2005, Serge Lang, Algebra, 3rd edition, Springer, →ISBN, page 3:

(algebra) The identity element, neutral element.

Let 𝐴 be an arbitrary integral domain. We say that an element 𝑎 ∈ 𝐴 is invertible or is a unit of 𝐴 if it has an inverse in 𝐴; in ℤ the units are ±1, […]

1990, A. I. Kostrikin, I. R. Shafarevich, editors, Algebra I, Springer, →ISBN, page 22:

(algebra) An element having an inverse, an invertible element; an associate of the unity.


known

From Middle English knowen, from Old English cnāwen (past participle).
Morphologically know + -n.

(algebra) A constant or variable the value of which is already determined.


meet

From Middle English meten, from Old English mētan (“to meet, find, encounter”), from Proto-West Germanic *mōtijan (“to meet”), from Proto-Germanic *mōtijaną (“to meet”), from Proto-Indo-European *meh₂d- (“to come, meet”).
Cognates
Cognate with Scots met, mete, meit (“to meet”), North Frisian mete (“to meet”), West Frisian mette, moetsje (“to meet”), Dutch ontmoeten (“to meet”), Low German möten (“to meet”), Danish møde (“to meet”), Norwegian Bokmål møte (“to meet”), Swedish möta (“to meet”), Icelandic mæta (“to meet”). Related to moot.

der

(algebra) The greatest lower bound, an operation between pairs of elements in a lattice, denoted by the symbol ∧.


central

en-adj

central (comparative more central, superlative most central)

more central (comparative)
most central (superlative)

Borrowed from Latin centrālis, from centrum (“centre”), from Ancient Greek κέντρον (kéntron); by surface analysis, centre + -al.

(algebra) Belong or relating to the center of an algebraic structure.

Of an element of an algebraic structure: which commutes with all other elements under multiplication

Of a unital algebra over a field: whose center is exactly equal to the image of the base field


ideal

From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea.
In mathematics, the noun ring theory sense was first introduced by German mathematician Richard Dedekind in his 1871 edition of a text on number theory. The concept was quickly expanded to ring theory and later generalised to order theory. The set theory and Lie theory senses can be regarded as applications of the order theory sense.

Let #92;mathbb#123;Z#125; be the ring of integers and let 2#92;mathbb#123;Z#125; be its ideal of even integers. Then the quotient ring #92;mathbb#123;Z#125;#47;2#92;mathbb#123;Z#125; is a Boolean ring.

The product of two ideals #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is an ideal #92;mathfrak#123;ab#125; which is a subset of the intersection of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125;. This should help to understand why maximal ideals are prime ideals. Likewise, the union of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is a subset of #92;mathfrak#123;a#43;b#125;.

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.

2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47:

If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.

2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45:

However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.

2010, W. D. Burgess, A. Lashgari, A. Mojiri, “Elements of Minimal Prime Ideals in General Rings”, in Sergio R. López-Permouth, Dinh Van Huynh, editors, Advances in Ring Theory, Springer (Birkhäuser), page 69:

(algebra, ring theory) A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.

理想

subring closed under multiplication by containing ring

1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119,
An ideal A of L is called complete if it contains all least upper bounds of its subsets that exist in L. Bishop and Schreiner [80] studied conditions under which joins of ideals in the lattices of all ideals and of all complete ideals coincide.

1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.

2011, George Grätzer, Lattice Theory: Foundation, Springer (Birkhäuser), page 125:

Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.

2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186:

partially ordered set
partially ordered set

w:Boolean prime ideal theorem#Prime ideal theorems

(algebra, order theory, lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).

If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.

1975, Zhe-Xian Wan, translated by Che-Young Lee, Lie Algebras, Pergamon Press, page 13:

What really put primitive ideals in enveloping algebras of semisimple Lie algebras on the map was Duflo's fundamental theorem that any such ideal is the annihilator of a very special kind of simple module, namely a highest weight module.

2006, W. McGovern, “The work of Anthony Joseph in classical representation theory”, in Anthony Joseph, Joseph Bernstein, Vladimir Hinich, Anna Melnikov, editors, Studies in Lie Theory: Dedicated to A. Joseph on His Sixtieth Birthday, Springer (Birkhäuser), page 3:

Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L#95;1#92;oplus#92;dots#92;oplusL#95;t of simple ideals (Theorem 5.2).

2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73:

(algebra, Lie theory) A Lie subalgebra (subspace that is closed under the Lie bracket) 𝖍 of a given Lie algebra 𝖌 such that the Lie bracket [𝖌,𝖍] is a subset of 𝖍.

The set of natural numbers with multiplication as the monoid operation (instead of addition) has multiplicative ideals, such as, for example, the set {1, 3, 9, 27, 81, ...}. If any member of it is multiplied by a number which is not a power of 3 then the result will not be a power of three.

(algebra) A subsemigroup with the property that if any semigroup element outside of it is added to any one of its members, the result must lie outside of it.


simple

simple (comparative simpler or more simple, superlative simplest or most simple)

simpler (comparative)
more simple (comparative)
simplest (superlative)
most simple (superlative)

Inherited from Middle English symple, simple, from Old French and French simple, from Latin simplex (“simple”, literally “onefold”) (as opposed to duplex (“double”, literally “twofold”)), from semel (“the same”) + plicō (“I fold”). See same and fold. Compare single, singular, simultaneous, etc.
Partially displaced native English onefold.

der

(heading, technical) Structurally uncomplicated.

(algebra, loosely, of an algebraic structure) Being non-trivial, and admitting no proper non-trivial quotients.

(group theory, of a group) Being non-trivial, and having no proper non-trivial normal subgroups (equivalently, no proper non-trivial quotient groups).

(module theory, of a module) Being non-trivial, and having no proper non-trivial submodules (equivalently, no proper non-trivial quotient modules).

(ring theory, of a ring) Being non-zero, and having no proper non-zero two-sided ideals (equivalently, no proper non-trivial quotient rings). For commutative rings, this definition coincides with that of a field.

equivalence relation
equivalence relation

(universal algebra, of an algebraic structure) Containing more than one element, and such that the only congruences on the structure are the diagonal relation (the equivalence relation a≡b⟺a=b) and the universal relation (the equivalence relation such that a≡b for all a,b). Equivalently, containing more than one element and having no proper non-trivial quotient algebras.

(category theory, of an object in a category with a terminal object) Being non-isomorphic to the terminal object, and such that its only quotient objects (up to isomorphism) are the terminal object and itself.

(algebra, of a Lie algebra) Being non-abelian and having no proper non-zero ideals. (Note that this is non-equivalent to the usual algebra sense; in particular, the abelian Lie algebra of dimension 1 over any given field is non-trivial and has no proper non-zero ideals, but is by convention not considered simple.)


complex

en-adj

complex (comparative more complex or (nonstandard) complexer, superlative most complex or (nonstandard) complexest)

more complex (comparative)
complexer (comparative nonstandard)
most complex (superlative)
complexest (nonstandard superlative)

From French complexe, from Latin complexus, past participle of complector (“I entwine, encircle, compass, infold”), from com- (“together”) and plectere (“to weave, braid”). May be analyzed as com- + -plex. See complect. Doublet of complexus.

complex polynomial

complex algebraic variety

(mathematics, algebra) Whose coefficients are complex numbers; defined over the field of complex numbers.


ring

From a shortening of German Zahlring (“number(s) ring”) (coined by German mathematician David Hilbert in 1892). Apparently first used in English in 1930, E. T. Bell, “Rings whose elements are ideals,” Bulletin of the American Mathematical Society.

The set of integers, #92;mathbb#123;Z#125;, is the prototypical ring.

multiplicative operation
multiplicative operation

(algebra) An algebraic structure which consists of a set with two binary operations: an additive operation and a multiplicative operation, such that the set is an abelian group under the additive operation, a monoid under the multiplicative operation, and such that the multiplicative operation is distributive with respect to the additive operation.

The definition of ring without unity allows, for instance, the set 2#92;mathbb#123;Z#125; of even integers to be a ring.

(algebra) An algebraic structure as above, but only required to be a semigroup under the multiplicative operation, that is, there need not be a multiplicative identity element.


split

en-verb

split (third-person singular simple present splits, present participle splitting, simple past and past participle split)

splits (present singular third-person)
splitting (participle present)
split (participle past)
split (past)

Attested since about 1567, from Middle Dutch splitten (“to split”) and/or Middle Low German splitten (“to split”), both ultimately from Proto-West Germanic *splittjan, an intensive form of Proto-West Germanic *splītan (“to split”), from Proto-Germanic *splītaną (whence Danish splitte, Low German splieten, German spleißen), from Proto-Indo-European *(s)pley- (“to split, splice”).
Compare Middle English ysplett (“split”, past participle of splatten (“to split”)), Old English speld (“splinter”), Old High German spaltan (“to split”), Old Irish sliss (“splinter”), Lithuanian spaliai (“flax sheaves”), Czech půl (“half”), Old Church Slavonic рас-плитати (ras-plitati, “to cleave, split”).

cog

past participle of <i class="Latn mention" lang="enm">splatten</i> (“to split”)

plah

divide along a more or less straight line

(algebra, transitive and intransitive, acts on a polynomial) To factor into linear factors.

(generally, of an algebraic structure) To be expressable as a direct sum of sub-modules, -algebras, etc.

acts on a polynomial; generally; generally

short exact sequence
short exact sequence

(of an object which expresses the relationship between algebraic structures, particularly a short exact sequence) To contain an object which may be so expressed.

acts on a polynomial; particularly a short exact sequence; particularly a short exact sequence

In the first case X²-2, the minimum polynomial of #92;sqrt 2, splits completely over #92;Q(#92;sqrt 2); in the second case we see that X³-2, the minimum polynomial of 3#92;sqrt 2, does not split completely over #92;Q(3#92;sqrt 2).

2007, John M. Howie, Fields and Galois Theory, Springer, page 103:


split

(algebra, of a short exact sequence) Having the middle object (group, module, etc.) equal to the direct sum of the others.


difference

en-noun

difference (countable and uncountable, plural differences)

From Middle English difference, from Old French difference, from Latin differentia (“difference”), from differēns (“different”), present participle of differre. Doublet of differentia. Morphologically differ + -ence.

know the difference between one's ass and a hole in the ground

know the difference between one's ass and one's elbow

(relational algebra) The set of elements that are in one set but not another ( scriptstyle A◌̅B).


depth

depth (countable and uncountable, plural depths)

From Middle English depthe, from Old English *dīepþ (“depth”), from Proto-Germanic *diupiþō (“depth”), equivalent to deep + -th (abstract nominal suffix). Cognate with Scots deepth (“depth”), Saterland Frisian Djüpte (“depth”), West Frisian djipte (“depth”), Dutch diepte (“depth”), Low German Deepde (“depth”), Danish dybde (“depth”), Icelandic dýpt (“depth”), Gothic 𐌳𐌹𐌿𐍀𐌹𐌸𐌰 (diupiþa, “depth”).

(algebra, ring theory) An invariant of rings and modules, encoding information about dimensionality; see Depth (ring theory).


flat

en-adj

flat (comparative flatter, superlative flattest)

flatter (comparative)
flattest (superlative)
flatt (alternative obsolete)
flatte (alternative obsolete)

bumpy

antonym(s) of “having no variations in altitude”

→ Portuguese: flat

From Middle English flat, a borrowing from Old Norse flatr (compare Norwegian and Swedish flat, Danish flad), from Proto-Germanic *flataz, from Proto-Indo-European *pleth₂- (“flat”); akin to Saterland Frisian flot (“smooth”), German Flöz (“a geological layer”), Ancient Greek πλατύς (platús), Latvian plats, Sanskrit प्रथस् (prathas, “extension”). Doublet of plat and pleyt.
The noun is from Middle English flat (“level piece of ground, flat edge of a weapon”), from the adjective.
The algebraic sense was coined by Serre in a 1956 paper, originally as French plat.

inh

(algebra, ring theory, of a ring homomorphism) Such that its target, regarded as a module over its source, is flat (as above).


join

joins (plural)
joyn (alternative obsolete)
joyne (alternative obsolete)
joyen (alternative obsolete)

From Middle English joinen, joynen, joignen, from Old French joindre, juindre, jungre, from Latin iungō (“join, yoke”, verb), from Proto-Indo-European *yewg- (“to join, unite”). Cognate with Old English iucian, iugian, ġeocian, ġyċċan (“to join; yoke”). More at yoke.

der

(algebra) The lowest upper bound, an operation between pairs of elements in a lattice, denoted by the symbol ∨.


field

→ Japanese: フィールド (fīrudo)

From Middle English feeld, feld, from Old English feld, from Proto-West Germanic *felþu, from Proto-Germanic *felþuz (“field”), ultimately from Proto-Indo-European *pleh₂- (“field, plain”) or *pleth₂- (“flat”) (with schwebeablaut).
Cognate with Scots feld, feild (“field”), North Frisian fjild (“field”), West Frisian fjild (“field”), Dutch veld (“field”), German Feld (“field”), Swedish fält (“field”). Related also to Old English folde (“earth, land, territory”), Old English folm (“palm of the hand”). More at fold.

veld

land area free of woodland, cities, and towns; open country

fushë

sports: area reserved for playing a game

Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain

Noetherian domain ⊂ integral domain ⊂ commutative ring

The set of rational numbers, #92;mathbb#123;Q#125;, is the prototypical field.

Whereas a ring has three binary operators: (1) an additive operator, (2) a subtractive operator, and (3) a multiplicative operator, a field has four binary operators: the three ring binary operators and (4) a divisive operator. (N.B.: Only the additive and multiplicative operators are axiomatic. The subtractive operator may be derived by combining the additive and the unary negative operators; the divisive operator may be derived by combining the multiplicative and the unary inversive operators.)

Any of various figurative meanings, often dead metaphors.

Any of certain structures serving cognition.

(algebra) A non-zero commutative ring in which all non-zero elements are invertible; a simple commutative ring.

veld

mathematics (algebra): a commutative ring satisfying the field axioms


content

en-noun

content (countable and uncountable, plural contents)

desctree

→ Russian: контент (kontent)
→ Armenian: կոնտենտ (kontent)

From Middle English content (plural contentes, contence), from Latin contentus, past participle of continēre (“to hold in, contain”), as Etymology 1, above. English apparently developed a substantive form of the adjective, which is not mirrored in Romance languages.

greatest common divisor
greatest common divisor

(algebra, ring theory, of a polynomial with coefficients in a GCD domain) The greatest common divisor of the coefficients; (of a polynomial with coefficients in an integral domain) the common factor of the coefficients which, when removed, leaves the adjusted coefficients with no common factor that is noninvertible.


localization

en-noun

localization (countable and uncountable, plural localizations)

localizations (plural)
localisation (alternative Commonwealth)

/ˌləʊkəlaɪˈzeɪʃən/

From localize + -ation; compare French localisation.

გაქართულება

specifically meaning localization to Georgian only

multiplicative inverse
multiplicative inverse

(algebra) A systematic method of adding multiplicative inverses to a ring.

lokalisasyon

3) Geometric interpretation of the localization.
 Let V be an irreducible algebraic variety. Then P = J(V) is a prime ideal of #92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93; and so #92;mathbb#123;C#125;#91;V#93;#61;#92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;#47;J(V) is an integral domain.
 The localization #92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;#95;P is a subring of #92;mathbb#123;C#125;(X#95;1,...,X#95;n) consisting of rational functions #92;#123;f#47;g#58;f,g#92;in#92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;,g#92;notinP#92;#125; which are defined on a nonempty subset of V. If V = {x} is a point, then P is maximal and #92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;#95;P#61;#92;#123;f#47;g#58;f,g#92;in#92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;,g(x)#92;ne 0#92;#125; consists of rational functions which are defined at x.

2007, Ivan Fesenko, “Rings and modules”, in G13ALS Algebra 2, 2007/2008 @ maths.nottingham.ac.uk, page 27:

(algebra) A ring of fractions of a given ring, such that the complement of the set of allowed denominators is an ideal.


localize

en-verb

localize (third-person singular simple present localizes, present participle localizing, simple past and past participle localized)

localizes (present singular third-person)
localizing (participle present)
localized (participle past)
localized (past)
localise (alternative)

From local + -ize. Compare French localiser.

(algebra, transitive) To produce (from a ring and an ideal in that ring) the ring of fractions, where the set of allowed denominators is the compliment of the given ideal.


kernel

→ Japanese: カーネル (kāneru)

From Middle English kernel, kirnel, kürnel, from Old English cyrnel, from Proto-West Germanic *kurnil, diminutive of Proto-Germanic *kurną (“seed, grain, corn”), equivalent to corn + -le. Cognate with Yiddish קערנדל (kerndl), Middle Dutch kernel, cornel, Middle High German kornel. Related also to Old Norse kjarni (“kernel”).

ядро

(mathematics, algebra) set of elements mapped to zero

algebraic structures
algebraic structure

multiplicative group
multiplicative group

(mathematics, algebra, loosely, of a morphism f between algebraic structures) The set of elements of the domain of f which are mapped to an identity element.

(mathematics, group theory, of a group homomorphism f:G→H between groups) The set of elements of G which are mapped to the identity of H (usually written as 0 in an additive group and as 1 in a multiplicative group).

(mathematics, ring theory, of a ring homomorphism f:R→S between rings) The set of elements of R which are mapped to the additive identity 0 of S.

linear transformation
linear transformation

(mathematics, linear algebra, of a linear transformation f:M→N between modules or vector spaces) The set of elements of M which are mapped to the additive identity 0 of N.

Meronyms: root, zero


order

en-noun

order (countable and uncountable, plural orders)

orders (plural)
ordre (alternative obsolete)

→ Swahili: oda

From Middle English ordre, from Old French ordre, ordne, ordene (“order, rank”), from Latin ōrdinem, accusative of ōrdō (“row, rank, regular arrangement”, literally “row of threads in a loom”), from Proto-Italic *ordō (“to arrange”), probably ultimately from Proto-Indo-European *h₂or-d-, from *h₂er-.
Related to Latin ōrdior (“begin”, literally “begin to weave”). In sense “request for purchase”, compare bespoke. Doublet of ordo.

cog

The monomial x²ʸ³ᶻ is of order 2#43;3#43;1#61;6.

(algebra, of a monomial) The sum of the exponents of the variables involved in the expression.

The quadratic polynomial ax²#43;bx#43;c, is said to be of order (or degree) 2 when a is nonzero.

(algebra, of a polynomial in one variable) The order of the leading monomial; (equivalently) the largest power of the variable involved in the given expression.


binary operation

en-noun

binary operation (plural binary operations)

The "four rules" of addition, subtraction, multiplication and division are examples of binary operations.

(algebra) Any operation taking two operands.


natural

en-adj

natural (comparative more natural, superlative most natural)

more natural (comparative)
most natural (superlative)
naturall (alternative obsolete)
nat'ral (alternative)

/ˈnæt͡ʃ(ə)ɹəl/

From Middle English natural, borrowed from Old French natural, naturel, from Latin nātūrālis, from nātus, the perfect participle of nāscor (“be born”, verb). Displaced native Old English ġecynde. By surface analysis, nature + -al.

luonnollinen

that exists and evolved within the confines of an ecosystem

(algebra) Closed under submodules, direct sums, and injective hulls.


degree

→ Burmese: ဒီဂရီ (diga.ri)

→ Juba Arabic: digrii

→ Swahili: digrii

→ Tagalog: digri

From Middle English degre, borrowed from Old French degré (French: degré), itself from Latin gradus, with the prefix de-.

Appendix:Roget MICRA thesaurus/Class I § 26. Degree

Appendix:Roget MICRA thesaurus/Class I § 31. Greatness

Appendix:Roget MICRA thesaurus/Class I § 32. Smallness

A quadratic polynomial is a polynomial of degree 2.

(algebra) The sum of the exponents of a term; the order of a polynomial.

степен

algebra: the maximal exponent of a polynomial

The set of complex numbers constitutes a field extension of degree 2 over the real numbers.

The Galois field #92;operatorname#123;GF#125;(125)#61;#92;operatorname#123;GF#125;(5³) has degree 3 over its subfield #92;operatorname#123;GF#125;(5).

(algebra, field theory) The dimensionality of a field extension.

степен

field theory: the dimensionality of a field extension


homogeneous

From Medieval Latin homogeneus, from Ancient Greek ὁμογενής (homogenḗs, “of the same race, family or kind”), from ὁμός (homós, “same”) + γένος (génos, “kind”). Compare homo- (“same”) and -ous (adjectival suffix).

der

The polynomial x²+5xy+y² is homogeneous of degree 2, because x², xy, and y² are all degree 2 monomials

(mathematics) In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).

Of polynomials, functions, equations, systems of equations, or linear maps:

(algebra, of a polynomial) Such that all its nonzero terms have the same degree.


characteristic

/ˌkæɹəktəˈɹɪstɪk/

[ˌkʰæɹəktəˈɹɪstɪk]

General American, Mary–marry–merry merger

/ˌkɛɹəktəˈɹɪstɪk/

From Ancient Greek χαρακτηριστικός (kharaktēristikós), from χαρακτηρίζω (kharaktērízō, “to designate by a characteristic mark”), from χαρακτήρ (kharaktḗr, “a mark, character”). By surface analysis, character + -istic.

χαρακτηριστικός

The characteristic of a field, if non-zero, must be a prime number.

1962 [John Wiley & Sons], Nathan Jacobson, Lie Algebras, 1979, Dover, page 289,
In this chapter we study the problem of classifying the finite-dimensional simple Lie algebras over an arbitrary field of characteristic 0.

1992, Simeon Ivanov (translator), P. M. Gudivok, E. Ya. Pogorilyak, On Modular Representations of Finite Groups over Integral Domains, Simeon Ivanov (editor), Galois Theory, Rings, Algebraic Groups and Their Applications, American Mathematical Society, page 87,
Let R be a Noetherian factorial ring of characteristic p which is not a field.

Traditionally, a complete, discretely valued field of characteristic zero, the maximal ideal of whose valuation ring is generated by the prime number p, has been called a p-adic field. In our terminology, the valuation ring of a p-adic field is a Cohen ring of characteristic zero whose residue field has characteristic p, and consequently a p-adic field is simply the quotient field of such a Cohen ring.

1993, S. Warner, Topological Rings, Elsevier (North-Holland), page 424:

multiplicative identity
multiplicative identity

(algebra, field theory, ring theory) For a given field or ring, a natural number that is either the smallest positive number n such that n instances of the multiplicative identity (1) summed together yield the additive identity (0) or, if no such number exists, the number 0.

field theory; ring theory; field theory; ring theory


center

centers (plural)
centre (alternative Australia Canada Commonwealth Ireland New-Zealand South-Africa UK including usually)

From Middle English center, centre, from Middle French centre, from Latin centrum, from Ancient Greek κέντρον (kéntron), from κεντεῖν (kenteîn, “to prick, goad”). Doublet of centrum.
* (group theory): In the notation Z(G), the letter Z derives from the German Zentrum (“center”).

sentrum

point equidistant from all points on the perimeter of a circle

مَرْكَز

point in the interior of figure with mean coordinates

(algebra, group theory, ring theory) The set of those elements (of a given algebraic structure) that commute with every other element, usually denoted Z(G). In the case of (semi-)groups, it is required that they commute under the (semi-)group operation; in the case of rings and (Lie) algebras, under multiplication (i.e. the bracket for Lie algebras)

centro

set of elements of a group that commute with all elements


algebra

en-noun

algebra (countable and uncountable, plural algebras)

⇒ Chinese: 代數 /代数 (dàishù)

→ Japanese: 代数

→ Korean: 대수 (daesu)

→ Vietnamese: đại số

/ˈæl.d͡ʒɪ.bɹə/

/ˈæl.d͡ʒə.bɹə/

Borrowed from Medieval Latin algebra, from the Arabic word الْجَبْر (al-jabr, “reunion, resetting of broken parts”) in the title of al-Khwarizmi's influential work الْكِتَاب الْمُخْتَصَر فِي حِسَاب الْجَبْر وَالْمُقَابَلَة (al-kitāb al-muḵtaṣar fī ḥisāb al-jabr wa-l-muqābala, “The Compendious Book on Calculation by Completion and Balancing”).

(countable, mathematics, algebra) Any of several objects of study in Algebra

An algebraic structure consisting of a module over a commutative ring (or a vector space over a field) along with an additional binary operation that is bilinear over module (or vector) addition and scalar multiplication.

алгебра

one of several other types of mathematical structure


binomial

Binominal nomenclature
Gerard of Cremona

Formed from Late Latin binōmium + -al. The derivation of binōmium is unclear. It was used by Gerard of Cremona in the 12th century. Suggested sources are the Latin nōmen (“name”), the Ancient Greek νομός (nomós, “distribution, pasture”), or the Old French nom (“name”).
Gérard de Crémone used the word in his translation of aν Arabic commentary on Euclid, corresponding to the Greek "εκ δυο ονοματων". Compare binomy and binominal, as well as the French binôme. By surface analysis, bi- + -nomial.

der

(algebra) A quantity expressed as the sum or difference of two terms.

binomi

algebra: quantity expressed as sum of two terms


associative

en-adj

associative (comparative more associative, superlative most associative)

more associative (comparative)
most associative (superlative)

Perhaps it is an advantage of the "associative algebraic geometry" we have tried to develop in foregoing chapters that it is independent of braidings and further generalizations because it will remain valid as long as the corresponding "function"-rings constructed in these theories are associative algebras.

2000, Freddy Van Oystaeyen, Algebraic Geometry for Associative Algebras, Marcel Dekker, page 235:

It is now generally accepted that the representation theory of associative algebras traces its origin to Hamilton's description of the complex numbers by pairs of real numbers.

2006, Ibrahim Assem, Daniel Simson, Andrzej Skowroński, Elements of the Representation Theory of Associative Algebras, 1: Techniques of Representation Theory, Cambridge University Press, page vii:

In this section we develop the basic theory of normed algebras, putting special emphasis on the case of complete normed unital associative complex algebras.

2014, Miguel Cabrera García, Ángel Rodríguez Palacios, Non-Associative Normed Algebras, Volume 1: The Vidav–Palmer and Gelfand-Naimark Theorems, Cambridge University Press, page 1:

(algebra, of a binary operator *) Such that, for any operands a,b and c, (a*b)*c=a*(b*c); (of a ring, etc.) whose multiplication operation is associative.


commutative

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

From French commuter (“to substitute or switch”) + -ative (“tending to”). See commute.

der

(algebra, of an algebraic structure) Having a commutative operation.


constant

From Middle English constant, from Old French constant, from Latin constantem, accusative of constans, from cōnstāre (“to stand firm”). Displaced native Old English singal.

We also establish that constants are local extremizers of the Tomas-Stein adjoint restriction inequality as well as of another inequality appearing in the program..

2015, Emanuel Carneiro, Damiano Foschi, Diogo Oliveira e Silva, Christoph Thiele, “A sharp trilinear inequality related to Fourier restriction on the circle”, in arXiv:

(algebra) A quantity that remains at a fixed value throughout a given discussion.


polynomial

/ˌpɒlɪˈnəʊmɪəl/

From poly- + -nomial, from νομός (nomós, “portion, part”), by analogy with binomial.

Unfortunately, an analogous attack on the octic polynomial becomes exceedingly complicated.

1999, William Dunham, Euler: The Master of Us All, Mathematical Association of America, published 1999, page 116:

Our upper bound is the best possible, and it implies the existence of low-rank factorizations of positive semidefinite bivariate matrix polynomials and representations of biforms as sums of few squares.

2016, Grigoriy Blekherman, Daniel Plaumann, Rainer Sinn, Cynthia Vinzant, “Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree”, in arXiv:

(algebra, strict sense) An 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_nxⁿ+a_n-1xⁿ⁻¹+...+a_0x⁰.


polynomial

/ˌpɒlɪˈnəʊmɪəl/

(algebra) Able to be described or limited by a polynomial.


unknown

From Middle English *unknowen, *uniknowen, uniknowe, from Old English unġecnāwen (“unknown”), equivalent to un- + known.

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

مُتَغَيِّر

algebra: a variable whose value is to be found


bracket

From earlier bragget, *bracket, from Middle English *braget, *braket (attested in braket nail), from Old French braguette (“the opening in the fore part of a pair of breeches, one's fly”), a diminutive of Old French brague (“knickers, britches”), from Old Occitan braga, from Latin brāca (“pants”), from Transalpine Gaulish *brāca, from Proto-Germanic *brāks, an early form of Proto-Germanic *brōks (“leggings, breeches, trousers”).

kllapë

generically any of “(”, “)”, “[”, “]”, “{”, “}”, “⟨”, “⟩”, “<”, “>”, etc.

(algebra) A pair of values that represent the smallest and largest elements of a range.


local

en-adj

local (comparative more local, superlative most local)

more local (comparative)
most local (superlative)

From Middle English local, from Late Latin locālis (“belonging to a place”), possibly also via Old French local; ultimately from Latin locus (“a place”).
The ring-theoretic senses derive from Krull, who first referred to Noetherian commutative rings with a unique maximal ideal as "Stellenring" (Stellen (“place”) + ring) in 1938. The term was inspired by algebraic geometry, where local rings encode information about the behavior of curves (surfaces, etc.) at points; hence, describe "local" behavior.

der

Flatness is a local property.

(mathematics, not comparable, of a condition or property) Detectable from the behavior of substructures understood to be "near points;" in particular:

(algebra, algebraic geometry, of a property P of a ring R (or an R-module M)) Such that the following conditions are equivalent: (1) P holds for R (M); (2) P holds for the localization R_p (M_p) for all prime ideals p of R; (3) P holds for the localization R_m (M_m) for all maximal ideals m of R.

(algebra, of a ring) Having a unique maximal (left) ideal.


lattice

→ Japanese: ラティス (ratisu)

From Middle English latis, from Middle French lattis (“lathing”), from Old French lattis, from latte (“a lath”), from Frankish *lattā, *lattu (“a lath”), from Proto-Germanic *lattǭ, *lattō, *laþþō (“board; plank; ledge”), from Proto-Indo-European *(s)latn-, *(s)lat- (“beam; log”). Cognate with Old High German latta (“lath”), (German Latte), Old English lætt (“lath”), Middle Low German lāde (“plank, counter, sales counter”), German Laden (“shop”). More at lath.

cog

(mathematics, algebra) 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.

(group theory) A discrete subgroup of Rⁿ which is isomorphic to Zⁿ (considered as an additive group) and which spans the real vector space Rⁿ.

(topology, Lie theory, generalizing sense 5.1) A discrete subgroup L of a given locally compact group G whose quotient space G/L has finite invariant measure.

Lie theory; generalizing sense 5.1; Lie theory; generalizing sense 5.1

(algebra, ring theory, generalizing sense 5.1, with respect to a vector space V over a field F which is the field of fractions of an integral domain R in F) A finitely generated R-submodule of V which spans V over F. (In this case the submodule is called an R-lattice'').

ring theory; generalizing sense 5.1; ring theory; generalizing sense 5.1


expansion

en-noun

expansion (countable and uncountable, plural expansions)

Borrowed from French expansion, from Latin expānsiō. By surface analysis, expand + -sion.

An act, process, or instance of expanding.

(algebra) The rewriting of an expression as a longer but equivalent sum of terms.


expand

en-verb

expand (third-person singular simple present expands, present participle expanding, simple past and past participle expanded)

expands (present singular third-person)
expanding (participle present)
expanded (participle past)
expanded (past)
expand (infinitive)
expand (first-person present singular)
expanded (first-person past singular)
expand (present second-person singular)
expandest (archaic present second-person singular)
expanded (past second-person singular)
expandedst (archaic past second-person singular)
expands (present singular third-person)
expandeth (archaic present singular third-person)
expanded (past singular third-person)
expand (plural present)
expanded (past plural)
expand (present subjunctive)
expanded (past subjunctive)
expand (imperative present)
expanding (participle present)
expanded (participle past)

contract

antonym(s) of “to change from a smaller form/size to a larger one”

contract

antonym(s) of “to increase the extent, number, volume or scope of”

factor

Recorded in Middle English since 1422 (as expanden, expaunden), from Anglo-Norman espaundre, from Latin expandere (“to spread out”), itself from ex- (“out, outwards”) + pandō (“to spread”). Doublet of spawn.

وَسَّعَ

(transitive) to increase the extent, number, volume or scope of

فَصَّلَ

(transitive) to express at length or in detail

разяснявам се

(intransitive) to speak or write at length or in detail

enlarge

Use the binomial theorem to expand #123;(x#43;1)#125;⁴.

(transitive, algebra) To rewrite (an expression) as a longer, yet equivalent, sum of terms.

разкривам скоби

(transitive, algebra) to rewrite as an equivalent sum of terms

The expression #123;(x#43;1)#125;⁴ expands to x⁴#43;4x³#43;6x²#43;4x#43;1.

(intransitive, algebra, of an expression) To become, by rewriting, a longer, yet equivalent, sum of terms.

sviluppare

(intransitive, algebra) to become an equivalent sum of terms


inversion

en-noun

inversion (countable and uncountable, plural inversions)

/ɪnˈvɜː.ʃən/

(algebra) An operation on a group, analogous to negation.


conjugate

The adjective (as “combined, united”) and noun are first attested in 1471, in Middle English, the verb in 1530; partly from Middle English conjugat(e) (“combined, united”), partly directly borrowed from New Latin coniugātus, the perfect passive participle of Latin coniugō (“to yoke together, combine; (New Latin) to conjugate, decline, inflect”) (see -ate (etymology 1, 2 and 3)), from con- (“with”) + iugō (“to join”). In Classical Latin, the word for conjugate (grammar) was dēclīnō, coniugō is a later back-formation from post-classical coniugātiō (“conjugation, declension”).

der

(algebra, of a complex number) A complex conjugate.

(algebra) More generally, any of a set of irrational or complex numbers that are zeros of the same polynomial with integral coefficients.

(algebra, field theory, of an element of an extension field) Given a field extension L / K and an element α ∈ L, any other element β ∈ L that is another root of the minimal polynomial of α over K.


monoid

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

Just like any abstraction, a monoid is useful to the extent we can write useful generic code assuming only the capabilities provided by the abstraction.

2023, Michael Pilquist, Rúnar Bjarnason, Paul Chiusano, Functional programming with Scala, Second edition, Shelter Island: Manning Publications Co, →ISBN:

functional programming
functional programming

(algebra, functional programming) A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.


identity

en-noun

identity (countable and uncountable, plural identities)

From Middle French identité, from Latin identitās.

(algebra, computing) Any function which maps all elements of its domain to themselves.

Zero is the identity for the addition of real numbers.

(algebra) An element of an algebraic structure which, when applied to another element under an operation in that structure, yields this second element.


additive identity

additive identity (plural additive identities)

(algebra) An element of an algebraic structure, generally denoted 0, which is an identity for an additive operation (generally denoted +). That is, x + 0 = 0 + x = x for any x in the structure.


multiplicative identity

multiplicative identity (plural multiplicative identities)

(algebra) An element of an algebraic structure, generally denoted 1, which is an identity for a multiplicative operation (generally denoted × or *, or by concatenation).


dependent variable

dependent variable (plural dependent variables)

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

In the equation z = x² + y, z is the dependent variable.

(algebra) In an equation, the variable whose value depends on one or more variables in the equation.


independent variable

independent variable (plural independent variables)

In the equation z = x² + y, x and y are the independent variables.

(algebra) In an equation, any variable whose value is not dependent on any other in the equation.


radical

PIE word
 *wréh₂ds
Inherited from Middle English radical, from Latin rādīcālis (“of or pertaining to the root, having roots, radical”).

radicale

ring theory: ideal of elements of a ring sharing a certain given property

radicale

ring theory: intersection of maximal submodules of a module

(algebra, commutative algebra, ring theory, of an ideal) Given an ideal I in a commutative ring R, another ideal, denoted Rad(I) or √, such that an element x ∈ R is in Rad(I) if, for some positive integer n, xⁿ ∈ I; equivalently, the intersection of all prime ideals containing I.

commutative algebra; ring theory; commutative algebra; ring theory

radicale

ring theory: intersection of prime ideals containing a given ideal

(algebra, ring theory, of a ring) Given a ring R, an ideal containing elements of R that share a property considered, in some sense, "not good".

(algebra, ring theory, of a module) The intersection of maximal submodules of a given module.


congruence

en-noun

congruence (countable and uncountable, plural congruences)

/ˈkɒŋ.ɡɹʊ.əns/

/ˈkɔŋ.ɡɹʉ.əns/

From Middle English congruence, from Latin congruentia (“agreement”), from congruēns, present active participle of congruō (“meet together, agree”).

համաձայնություն

the quality of agreeing or corresponding

equivalence relation
equivalence relation#English

algebraic structure
algebraic structure#English

(universal algebra) Any equivalence relation defined on an algebraic structure which is preserved by operations defined by the structure.

kongruenssi

algebra, equivalence relation preserved by operations


discriminant

(algebra) An expression that gives information about the roots of a polynomial; for example, the expression D = b² - 4ac determines whether the roots of the quadratic equation ax² + bx + c = 0 are real and distinct (D > 0), real and equal (D = 0) or complex (D < 0).


section

From Middle English seccioun, from Old French section, from Latin sectiō (“cutting, cutting off, excision, amputation of diseased parts of the body, etc.”), from sectus, past participle of secāre (“to cut”). More at saw.

разрэ́з

surgery: incision or the act of making an incision

A cross-section (image that shows an object as if cut along a plane).

(generalizing the topology sense, algebra, category theory) A right inverse of a morphism in some category


integral

en-adj

integral (comparative more integral, superlative most integral)

more integral (comparative)
most integral (superlative)

Borrowed from Middle French integral, from Medieval Latin integrālis, from Latin integer (“entire”); see integer.

(algebra, commutative algebra, of a ring element in a ring B relative to a subring A) Being the root of some monic polynomial in A.


normal

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(algebra, of a subgroup) With cosets which form a group.

(algebra, of a field extension of a field K) Which is the splitting field of a family of polynomials in K.


rank

en-noun

rank (countable and uncountable, plural ranks)

From Middle English rank (“line, row”), from Old French ranc, rang, reng (“line, row, rank”) (Modern French rang), from Frankish *hring (“ring”), from Proto-Germanic *hringaz (“something bent or curved”).
Akin to Old High German (h)ring, Old Frisian hring, Old English hring, hrincg (“ring”), Old Norse hringr (“ring, circle, queue, sword; ship”). Doublet of ring and rink.

cog

(algebra) The maximum quantity of D-linearly independent elements of a module (over an integral domain D).


selection

en-noun

selection (countable and uncountable, plural selections)

From Latin sēlēctiō (“the act of choosing out, selection”), from sēlēctus, perfect passive participle of sēligō (“choose out, select”), from sē- (“apart”) + legō (“gather, select”).

der

(algebra) A unary operation that denotes a subset of a relation.


concomitant

First attested 1607; from Middle French concomitant, from Latin concomitāns, the present participle of concomitor (“I accompany”), from con- (“together”) + comitor (“I accompany”), from comes (“companion”).

(algebra) An invariant homogeneous polynomial in the coefficients of a form, a covariant variable, and a contravariant variable.


rack

From Middle English rakke, rekke, from Middle Dutch rac, recke, rec (Dutch rek), see rekken.

(algebra) A set with a distributive binary operation whose result is unique.


correlation

en-noun

correlation (countable and uncountable, plural correlations)

/ˌkɒɹ.əˈleɪ.ʃən/

/ˌkɔɹ.əˈleɪ.ʃən/

/ˌkɑ.ɹəˈleɪ.ʃən/

From Middle French corrélation. Morphologically correlate + -ion.

(algebra) An isomorphism from a projective space to the dual of a projective space, often to the dual of itself.


identity element

identity element (plural identity elements)

Therefore the number 0 is not considered an identity element for subtraction, even though x-0#61;x for all x, since 0-x#92;nex.

1990, Daniel M. Fendel, Diane Resek, Foundations of Higher Mathematics, Volume 1, Addison-Wesley, page 269:

Proof. If e and e' are both identity elements, then we have ee'=e since e' is an identity element, and ee'=e' since e is an identity element. Thus
e=ee'=e'.

2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17, Let (G,·) be a group. Then the identity element e∈G is unique. […]

Sometimes, to avoid ambiguity, we may use the notation e#95;M for the identity element of M.
If multiplicative notation is used then we use the term identity element, and often use the notation 1, or 1#95;M, for the neutral element e.

2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 41:

(algebra) An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element.

identity

element that when applied with a binary operation leaves any other element unchanged


left identity

(algebra) An element of a structure that produces an identity mapping when applied on the left.

elemento neutro por la izquierda

element that produces an identity mapping when applied on the left


right identity

right identity (plural right identities)

(algebra) An element of a structure that produces an identity mapping when applied on the right.

elemento neutro por la derecha

element that produces an identity mapping when applied on the right


prime

en-adj

prime (comparative primer, superlative primest)

primer (comparative)
primest (superlative)

From Middle English prime, from Old French prime and its etymon, Latin prīmus (“first”), from earlier prīsmos < *prīsemos < Proto-Italic *priisemos, ultimately from Proto-Indo-European *per- (“beyond, before”). Doublet of primo and primus.
The noun sense "apostrophe-like symbol" originates from the fact that the symbol ′ was originally a superscript Roman numeral one.

der

(algebra, of an ideal) Having its complement closed under multiplication.

素

mathematics: having its complement closed under multiplication

(algebra, of a nonzero module) Such that the annihilator of any nonzero submodule is equal to the annihilator of the whole module.


prime

composite

Some poems, echoing the purpose of early poetic treatises on scientific principles, attempt to elucidate the mathematical concepts that underlie prime numbers. Others play with primes’ cultural associations. Still others derive their structure from mathematical patterns involving primes.

2013 July-August, Sarah Glaz, “Ode to Prime Numbers”, in American Scientist, volume 101, number 4:

3 is a prime.

mathematical structure
mathematical structure

(algebra, number theory) A prime element of a mathematical structure, particularly a prime number.

просто число

algebra: prime element of a mathematical structure


configuration

en-noun

configuration (countable and uncountable, plural configurations)

/kənˌfɪɡ.əˈɹeɪ̯.ʃən/

/kənˌfɪɡ.jəˈɹeɪ̯.ʃən/

/kənˌfɪɡ.əˈɹæɪ̯.ʃən/

From Middle French configuration, from Latin cōnfigūrātiō.
Morphologically configure + -ation

määritys

computing: process of customizing a computing environment

(countable, algebra) A finite set of points and lines (and sometimes planes), generally with equal numbers of points per line and equal numbers of lines per point.


variety

variety (countable and uncountable, plural varieties)

varieties (plural)
variete (alternative)
varietie (alternative)
variëty (alternative)

From Middle French varieté (“variety”) (modern French variété (“variety; genre, type”)) or directly from its etymon Latin varietās (“difference; diversity, variety”) + English -ty (suffix forming abstract nouns from adjectives); by surface analysis, various + -ety. Varietās is derived from varius (“different, diverse, various; variegated”) (possibly ultimately from Proto-Indo-European *h₁weh₂- (“to abandon; to give out; to leave”)) + -tās (suffix forming feminine abstract nouns indicating a state of being). The English word displaced the native Old English mislīcnes.
Sense 1.3.2 (“total number of distinct states of a system; logarithm to the base 2 of the total number of distinct states of a system”) was coined by the English psychiatrist William Ross Ashby (1903–1972) in his work An Introduction to Cybernetics (1956).
Cognates
* Galician variedade (“variety”)
* Italian varietà (“difference; variety”)
* Portuguese variedade (“variety”)
* Spanish variedad (“breed; variety”)

der

A collection or number of different things.

In universal algebra: an equational class; the class of all algebraic structures of a given signature, satisfying a given set of identities.

varisto

in universal algebra: the class of all algebraic structures of a given signature, satisfying a given set of identities

(algebraic geometry) Ellipsis of algebraic variety (“the set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition”).

algebraic variety
algebraic variety#English


module

Borrowed from French module, from Latin modulus (“a small measure, a measure, mode, meter”), diminutive of modus (“measure”) (whence mode). Doublet of modulus and mold.

Meronym: element

Modules over a ring are a generalization of abelian groups (which are modules over #92;textstyle#92;mathbb#123;Z#125;).

1974, Thomas W. Hungerford, Algebra, Springer, page 168:

Approximately forty-five years ago K. Morita presented the first major results on equivalences and dualities between categories of modules over a pair of rings.

2004, Robert R. Colby, Kent R. Fuller, Equivalence and Duality for Module Categories (with Tilting and Cotilting for Rings), Cambridge University Press, page vii:

One defines in like manner right K-modules and two-sided K-modules. If K is commutative, then every left K-module is automatically equipped with the structure of right and a two-sided K-module.

2012, A. A. Kirillov, Elements of the Theory of Representations, Springer, page 29:

scalar multiplication
scalar multiplication

(algebra, ring theory) An abelian group equipped with the operation of multiplication by an element of a ring (or another of certain algebraic objects), representing a generalisation of the concept of vector space with scalar multiplication.

模

type of algebraic structure, generalization of vector spaces over non-field rings


norm

From Latin norma (“a carpenter's square, a rule, a pattern, a precept”). Doublet of norma.

норма

math: function that maps vectors to non-negative reals

A quaternion algebra (a,b) over ksplits if and only if b is a norm from the field extension k(#92;sqrt#123;a#125;)#47;k, i.e. if and only if there is some x in k(#92;sqrt#123;a#125;) which has field norm exactly equal to b.

(mathematics) Any of several generalizations of the above: a field norm, ideal norm, etc.

(algebra) An element of the image of some (generalized) norm, the element then said to be from the norm in question, or from the structure which gave rise to the norm.


embedded

en-adj

embedded (comparative more embedded, superlative most embedded)

more embedded (comparative)
most embedded (superlative)

(algebra, of an associated prime of a module) Not minimal with respect to inclusion (among associated primes), i.e. having a proper sub-ideal which is an associated prime.


associate

From the substantivization of the above adjective, see -ate (noun-forming suffix) and Etymology 1 for more.

(algebra) One of a pair of elements of an integral domain (or a ring) such that the two elements are divisible by each other (or, equivalently, such that each one can be expressed as the product of the other with a unit).


coherent

coherent (comparative more coherent, superlative most coherent)

more coherent (comparative)
most coherent (superlative)
cohærent (alternative)

/kəʊˈhɪə.ɹənt/

/kəʉˈhɪə.ɹənt/

From Middle French coherent, from Latin cohaērēns, from co- + haereō. By surface analysis, cohere + -ent.

(algebra, of a module) Finitely generated and such that all finitely generated submodules are finitely presented.

(algebra, of a ring) Such that every finitely generated (left) ideal is finitely presented.


vector space

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A vector space is a set of vectors which can be linearly combined.

Each vector space has a basis and dimension.

(algebra, geometry, topology) A set of elements called vectors, together with some field and operations called addition (mapping two vectors to a vector) and scalar multiplication (mapping a vector and an element in the field to a vector), satisfying a list of constraints; equivalently, a module over a field.


effective

effective (comparative more effective, superlative most effective)

more effective (comparative)
most effective (superlative)
eff. (alternative)

From French effectif, from Latin effectīvus (“productive; effective”), from efficiō (“I make; I bring about”), equivalent to effect + -ive.

der

(algebra, of a group action) Such that no group element acts trivially.


rig

From ring (“algebraic structure”), omitting the letter n to suggest the lack of negatives. Compare rng (“structure like a ring but lacking a multiplicative identity”).

The set of natural numbers N with the usual operations of addition and multiplication is a rig, but not a ring. The set of integers Z is a ring. For a rig/ring (R,0,+,1,−), the set of polynomials R[x] on a generator x with the usual operations of addition and multiplication is also a rig/ring.

2004, ACM SIGPLAN Notices, volume 39, ACM Press, page 81:

2004, Jerzy Marcinkowski (editor), Computer Science Logic: 18th International Workshop, CSL 2004, Proceedings, Springer, LNCS 3210, page 17,
It follows that for each object A its endomorphisms End_C(A) = C(A,A) has the structure of what is now called a rig, that is to say a (commutative) ring without negatives.

(algebra, ring theory) An algebraic structure similar to a ring, but without the requirement that every element have an additive inverse.

semiring

algebraic structure like a ring but without additive inverses


transpose

en-verb

transpose (third-person singular simple present transposes, present participle transposing, simple past and past participle transposed)

transposes (present singular third-person)
transposing (participle present)
transposed (participle past)
transposed (past)

From Middle English transposen, from Old French transposer, from Latin trānspositus, perfect passive participle of trānspōnō (“to put across”), from trāns (“across”) + pōnō (“to put”).

(transitive, algebra) To move (a term) from one side of an algebraic equation to the other, reversing the sign of the term.

пренасям

algebra: to move (a term) to the other side of an equation


valuation

en-noun

valuation (countable and uncountable, plural valuations)

From Middle French valuation, noun of action from valuer, from Old French valoir.

(algebra) A measure of size or multiplicity.


assassin

Assassin (disambiguation)
Nicolas Bourbaki

From either French assassin or Italian assassino, from Arabic أَسَاسِيِّين (ʔasāsiyyīn, “people who are faithful to the foundation [of the faith]”) and the folkloric etymology Arabic حَشَّاشِين (ḥaššāšīn, “hashish users; low-lives”).
The mathematical sense was introduced by Bourbaki, playing on the notation operatorname Ass(M) and the fact that an associated prime is the annihilator of an element, and so is said to kill that element.

der

أَسَاسِيِّين

people who are faithful to the foundation 􂀿of the faith􂁀

(algebra, somewhat humorous) An associated prime of a module.


trivial

en-adj

trivial (comparative more trivial, superlative most trivial)

more trivial (comparative)
most trivial (superlative)
triviall (alternative obsolete)

PIE word
 *tréyes
* From Latin triviālis (“appropriate to the street-corner, commonplace, vulgar”), from trivium (“place where three roads meet”). Compare trivium, trivia.
* From the distinction between trivium (“the lower division of the liberal arts; grammar, logic and rhetoric”) and quadrivium (“the higher division of the seven liberal arts in the Middle Ages, composed of geometry, astronomy, arithmetic, and music”).

трывія́льны

(mathematics) of being the simplest possible case

(mathematics) Of, relating to, or being the simplest possible case.

(algebra, of an algebraic structure or ideal thereof) Containing only one element; having an underlying set which is a singleton.


congruent

en-adj

congruent (comparative more congruent, superlative most congruent)

more congruent (comparative)
most congruent (superlative)

/ˈkɔŋ.ɡɹʉ.ənt/

From Middle English congruent, from Latin congruēns, present active participle of congruō (“meet together, agree”).

(algebra) Satisfying a congruence relation.


virtually

en-adv

virtually (comparative more virtually, superlative most virtually)

more virtually (comparative)
most virtually (superlative)

/ˈvɜːt͡ʃʊəli/

/ˈvɜːt͡ʃəli/

/ˈvɜːt͡ʃʉəli/

/ˈvɵːt͡ʃʉəli/

/ˈvɵːt͡ʃəli/

From Middle English vertually; equivalent to virtual + -ly.

In this case W has a diagram with 2 vertices and one edge labelled ∞, and is therefore the infinite dihedral group. Thus W is virtually ℤ.

1998, Daryl Cooper, D. D. Long, A. W. Reid, “Infinite Coxeter Groups are Virtually Indicable”, in Proceedings of the Edinburgh Mathematical Society, volume 41, page 309:

In this paper, we are mainly concerned with homogeneity in the class of finitely generated virtually free groups, i.e. finitely generated groups with a free subgroup of finite index.

2018, Simon André, “Virtually free groups are almost homogeneous”, in arXiv, section 1:

(algebra) Of a subgroup of finite index.


group theory

en-noun

group theory (usually uncountable, plural group theories)

group theories (plural)
theory of groups (alternative)

THE feeling seems to be developing among physicists that group theory can be used as and when required, as many other techniques already are, without its being necessary for everyone to study all the whys and wherefores of the mathematical details involved.

1968, Arthur P. Cracknell, editor, Applied Group Theory: Selected Readings in Physics, Pergamon Press, page ix:

1995, Jean Brocas, Chapter 4: The Use of Group Theory in the Study of Non-Rigid Molecules, Danail Bonchev, D. H. Rouvray (editors), Chemical Group Theory: Techniques and Applications, Gordon and Breach, page 133,
Planar cis and/or trans internal rotation tunnelling have been discussed in terms of group theory, leading to theoretical spectra which have been compared to experimental work.

(algebra, uncountable) The mathematical theory of groups.


loop

From Middle English loupe (“noose, loop”), earlier lowp-knot (“loop-knot”), of North Germanic origin, from Old Norse hlaup (“a run”), used in the sense of a "running knot", from hlaupa (“to leap”), ultimately from Proto-Germanic *hlaupaną (“to leap, run”). Compare Swedish löp-knut (“loop-knot”), Danish løb-knude (“a running knot”), Danish løb (“a course”). More at leap. The verb is derived from the noun.

der

(algebra) A quasigroup with an identity element.


transcendental

en-adj

transcendental (comparative more transcendental, superlative most transcendental)

more transcendental (comparative)
most transcendental (superlative)

algebraic

antonym(s) of “not the root of a polynomial with rational coefficients”

algebraic

antonym(s) of “containing elements that are not the root of a polynomial”

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The theory of transcendental numbers was originated by Liouville in his famous memoir^† of 1844 in which he obtained, for the first time, a class, très-étendue, as it was described in the title of the paper, of numbers that satisfy no algebraic equation with integer coefficients.

1975, Alan Baker, Transcendental Number Theory, 2nd edition, Cambridge University Press, published 1990, page 1:

If the distribution of decimal digits of #92;pi (or any other transcendental number) is truly random (suspected but not yet mathematically proven!), given any arbitrary finite sequence of whole numbers, that sequence would be included an infinite number of times in the decimal expansion of #92;pi.

2005, Juan G. Roederer, Information and Its Role in Nature, Springer, page 28:

(algebra, number theory, field theory, of a number or an element of an extension field) Not algebraic (i.e., not the root of any polynomial that has positive degree and rational coefficients).

irrational

2006, Steven Roman, Field Theory, Springer, 2nd Edition, Graduate Texts in Mathematics 158, page 108,
Suppose that F<E is purely transcendental. Show that any simple extension of F contained in E (but not equal to F) is transcendental over F.

(algebra, field theory, of an extension field) That contains elements that are not algebraic.

transsendenttinen

not algebraic; containing elements that are not algebraic


algebraic number

en-noun

algebraic number (plural algebraic numbers)

The golden ratio (φ) is an algebraic number since it is a solution of the quadratic equation x²-x-1=0, whose coefficients are integers.

The square root of a rational number, √, is an algebraic number since it is a solution of the quadratic equation nx²-m=0, whose coefficients are integers.

Thus, the equation x-eʸ#61;0 is satisfied for x#61;1,#92;y#61;0 and for no other pair of algebraic numbers.

1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435:

As a matter of fact, it is not true that the algebraic numbers above can be factored uniquely, but the first case of failure occurs when p = 23.

1921, L. J. Mordell, Three Lectures on Fermat's Last Theorem, page 16:

(i) It is easily checked that the set of all algebraic numbers is countable, whereas the set of all complex numbers is uncountable (this non-constructive proof goes back to Cantor).

1991, P. M. Cohn, Algebraic Numbers and Algebraic Functions, Chapman & Hall, page 83, The existence of such 'transcendental' numbers is well known and it can be proved at three levels

(algebra, number theory) A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.

代數數 /代数数

element of a number field that is a root of a polynomial with integer coefficients


polynomial equation

polynomial equation (plural polynomial equations)

(algebra) Any algebraic equation in which one or both sides are in the form of a polynomial.


algebraic equation

algebraic equation (plural algebraic equations)

(algebra) A mathematical equation in which one or both sides is an algebraic expression, such as 2x + 7y = 3.


algebraic

en-adj

algebraic (comparative more algebraic, superlative most algebraic)

more algebraic (comparative)
most algebraic (superlative)

transcendental

antonym(s) of “that is the root of some polynomial”

transcendental

antonym(s) of “whose every element is the root of some polynomial”

/ˌæl.d͡ʒɪˈbɹeɪ.ɪk/

/ˌæl.d͡ʒɪˈbɹæɪ.ɪk/

Partly from algebra + -ic and partly from French algébraïque.

(algebra, number theory, of a number) Which is a root of some polynomial whose coefficients are rational.

(algebra, of a field) Whose every element is a root of some polynomial whose coefficients are rational.


adjoin

en-verb

adjoin (third-person singular simple present adjoins, present participle adjoining, simple past and past participle adjoined)

adjoins (present singular third-person)
adjoining (participle present)
adjoined (participle past)
adjoined (past)

From Anglo-Norman adjoindre, from Latin adiungō.

The ring adjunction #92;mathbb#123;Q#125;#91;#92;sqrt#123;2#125;#93; can be obtained from #92;mathbb#123;Q#125; by adjoining #92;sqrt#123;2#125; to #92;mathbb#123;Q#125;.

The field adjunction #92;mathbb#123;Q#125;(#92;pi) can be obtained from #92;mathbb#123;Q#125; by adjoining #92;pi to #92;mathbb#123;Q#125;.

(transitive, mathematics, chiefly algebra and number theory) To extend an algebraic object (e.g. a field, a ring, etc.) by adding to it an element not belonging to it and then ensuring closure, for example by also adding all finite power series of the element.

adjoindre

mathematics: to add (an element) to a structure


uncia

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From Latin uncia (“various Roman units”). Doublet of ounce, inch, onça, onza, oka, ouguiya, and awqiyyah.

doublet

(algebra, obsolete) A numerical coefficient in a binomial.


divisor

делител

An integer that divides another integer an integral number of times, the former being called a divisor of the latter.

(algebra, more generally) A ring element which can be multiplied (by some other ring element) to yield a third ring element, the first being called a divisor of the third. If the ring is noncommutative, then one specifies whether a divisor is left, right, or two-sided.


CSA

en-noun

CSA (countable and uncountable, plural CSAs)

Cartan subalgebra
Cartan subalgebra#English

(algebra, Lie theory) Initialism of Cartan subalgebra.

central simple algebra
central simple algebra#English

(algebra, ring theory) Initialism of central simple algebra.


ACC

ACC (countable and uncountable, plural ACCs)

ascending chain condition
ascending chain condition#English

(algebra) Initialism of ascending chain condition.


SU

special unitary group
special unitary group#English

(algebra) Abbreviation of special unitary group.


STA

(algebra) Abbreviation of spacetime algebra (in mathematical physics).


indeterminate

1862, H. J. Stephen Smith, Report on the Theory of Numbers—Part III, Report of the 31st Meeting of the British Association for the Advancement of Science, British Association for the Advancement of Science, page 292,
The form is linear, quadratic, cubic, biquadratic or quartic, quintic, &c., according to its order in respect of the indeterminates it contains; and binary, ternary, quaternary, &c., according to the number of its indeterminates. Thus x²+y² is a binary quadratic form, x³+y³+z³-3xyz is a ternary cubic form.

1892, Henry B. Fine, Kronecker and His Arithmetical Theory of the Algebraic Equation, Thomas S. Fiske, Harold Jacoby (editors), Bulletin of the New York Mathematical Society, Volume 1, New York Mathematical Society, page 179,
Such a factor is therefore an integral function of x and the indeterminates u_1,u_2,…u_n with coefficients belonging to the domain of rationality (R',R,..) and may be represented by g(x,u_1,u_2,..u_n).

2006, Alexander B. Levin, Difference Algebra, M. Hazewinkel, Handbook of Algebra, page 251,
Let T=T_σ and let S be the polynomial R-algebra in the set of indeterminates y_i,τ_(i∈I,τ∈T) with indices from the set I×T.

(algebra, strict sense) A symbol that resembles a variable or parameter but is used purely formally and neither signifies nor is ever assigned a particular value;

A symbol that resembles a variable or parameter but is used purely formally and neither signifies nor is ever assigned a particular value;
(loose sense) a variable.


socle

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From French socle, from Italian zoccolo (“wooden shoe”), from Latin socculus, diminutive of soccus (“sock”). Doublet of zoccolo.

(algebra, ring theory) The sum of the minimal normal submodules of a given R-module of a given ring R.


isolated

en-adj

isolated (comparative more isolated, superlative most isolated)

more isolated (comparative)
most isolated (superlative)

A rendering into English of French isolé.

(algebra, of an associated prime of a module) Minimal with respect to inclusion (among associated primes).


algebraic integer

en-noun

algebraic integer (plural algebraic integers)

A Gaussian integer z=a+ib is an algebraic integer since it is a solution of either the equation z²+(-2a)z+(a²+b²)=0 or the equation z-a=0.

An algebraic number is said to be an algebraic integer if the coefficient of the highest power of x in the minimal polynomial P is 1. The algebraic integers in an algebraic number field k form a ring R.

1984, Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, page 62:

1989, Heinrich Rolletschek, Shortest Division Chains in Imaginary Quadratic Number Fields, Patrizia Gianni (editor), Symbolic and Algebraic Computation: International Symposium, Springer, LNCS 358, page 231,
Let O_d be the set of algebraic integers in an imaginary quadratic number field Q [√],d<0, where d is the discriminant of O_d.

We consider the problem of finding all algebraic integers which belong to a bounded subset of the complex plane together with their conjugates.

2010, Pierre Moussa, “Localisation of algebraic integers and polynomial iteration”, in Sergiy Kolyada, Yuri Nanin, Martin Möller, Pieter Moree, Thomas Ward, editors, Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, American Mathematical Society, page 83:

(algebra, number theory) A real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers; equivalently, an algebraic number whose minimal polynomial (lowest-degree polynomial of which it is a root and whose leading coefficient is 1) has integer coefficients.

代數整數 /代数整数

algebraic number whose minimal polynomial has integer coefficients


affected

en-adj

affected (comparative more affected, superlative most affected)

more affected (comparative)
most affected (superlative)

an affected equation


DVR

discrete valuation ring
discrete valuation ring#English

(algebra) Initialism of discrete valuation ring.


DCC

en-noun

DCC (countable and uncountable, plural DCCs)

descending chain condition
descending chain condition#English

(algebra) Initialism of descending chain condition.


toric

en-adj

toric (comparative more toric, superlative most toric)

more toric (comparative)
most toric (superlative)

Pertaining to or shaped like a torus, or a section of a torus; toroidal.

(geometry and algebra) Which, in any of several technical senses, admits a high degree of symmetry, allowing combinatorial methods to be used in its study.

(algebraic geometry, of a variety) Containing an algebraic torus as a dense subset, such that the group action of the torus on itself extends to the whole space; or, the embedding map taking the torus into the space. See Toric variety on Wikipedia.Wikipedia

(geometry, of a manifold, generalizing the case of toric varieties) (Narrowly) A compact smooth toric variety. (Broadly) Quasitoric: a closed, real, even-dimensional smooth manifold equipped with an effective, smooth action by an algebraic torus whose orbits are simple complex polytopes and such that the action is locally the same as a faithful real representation of the group.

(algebraic geometry, of a stack) Any of several generalizations of the notion of toric varieties to stacks: the stack quotient of a toric variety by its torus; the stack quotient of a toric variety by a subgroup of its torus.

(commutative algebra, of an ideal) Generated by differences of monomials.


superellipse

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(algebra, geometry) a geometric shape constituting a transition between a rectangle and a circle; a closed curve, of which the circle and ellipse are special cases, whose parametric equation is x = a.cos^(2/r)t, y = b.cos^(2/r)t


superellipsoid

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(algebra, geometry) A closed surface, of which the sphere and ellipsoid are special cases, whose parametric equation is x = a.cos^(2/r)t, y = b.cos^(2/r)t, z = c.cos^(2/r)t.


separable

separable (comparative more separable, superlative most separable)

more separable (comparative)
most separable (superlative)

From Middle English separable, from Middle French separable and its etymon Latin sēparābilis.

x²#43;1 is separable, since its roots are i and -i.

(mathematics, algebra) Any of several technical senses relating to the behavior of polynomials or objects over which polynomials can be defined:

(of a polynomial) Having no repeated roots (where roots are considered in an algebraic closure)

(Galois theory, obsolete, of a polynomial P) Such that none of the irreducible factors of P have a repeated root.

(Galois theory, of an algebraic field extension E/F) Such that the minimal polynomial of every element of E is a separable polynomial.

(abstract algebra, of an algebra over a ring) Satisfying any of several technical conditions on the center of the algebra which generalize the situation of field extensions; see Separable algebra on Wikipedia.Wikipedia


irreducible

irreducible (comparative more irreducible, superlative most irreducible)

more irreducible (comparative)
most irreducible (superlative)

/ˌɪɹɪˈdjuːsɪbəl/

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(algebra, of an element of a ring) Whose only divisors are units and associates.


difference of two squares

en-noun

difference of two squares (plural differences of two squares)

The fraction #92;frac#123;x-8#125;#123;x²-64#125; can be simplified to #92;frac#123;1#125;#123;x#43;8#125; using the difference of two squares.

(algebra) The identity x²-y²=(x+y)(x-y), that holds for all real numbers.


projective

en-adj

projective (comparative more projective, superlative most projective)

more projective (comparative)
most projective (superlative)

Formed by the suffixation of prōiect- (the perfect passive participial stem of the Classical Latin prōiciō, whence the English verb project) with the English -ive, forming project + -ive; however, compare the post-Classical Latin prōiectīvus (“relating to purging”).

cog

(algebra) In a technical sense, general (but not necessarily so general as to be free); involving such objects:

(module theory, of a (left) module M over a ring R) Such that there is another (left) R-module N with M⊕N a free R-module. Equivalently, projective in the category of modules (see below).

(category theory, most generally, of an object P in a category) Such that, given an epimorphism e:E→X and morphism f:P→X, f factors through e; that is, there exists a morphism ̃f:P→E with e∘̃f=f.

(group theory, of a profinite group G) Such that every epimorphism from a profinite group onto G has a right inverse which is a homomorphism.

(homological algebra, of a resolution) Such that every object in the resolution is projective.


quadratic equation

en-noun

quadratic equation (plural quadratic equations)

When I was seven years old at a new school, I cried all day, every day. When I was 14 years old, I cried in maths lessons. An improvement, I guess. It wasn’t that I was bad at maths; I knew my way around a quadratic equation, let me tell you.

2015 September 8, Katie Everson, “Why I've written a book for teenagers about taking drugs”, in The Guardian:

The arid argot of the estate agent’s schedule becomes obsolete when the task is to sell a pub-restaurant at the edge of a Highland wilderness. It’s like trying to describe the glory of the Lake District with a quadratic equation.

2018 November 25, Kevin McKenna, “Remote island pub for sale ... just don’t expect a quiet life”, in The Guardian:

The quadratic equation has frustrated math students for millenniums. […] If, while watching the Super Bowl, you had wanted to estimate how far a pass thrown by Patrick Mahomes traveled through the air, you would have been solving a quadratic equation.

2020 February 5, Kenneth Chang, Jonathan Corum, “This Professor’s ‘Amazing’ Trick Makes Quadratic Equations Easier”, in The New York Times:

If there’s one skill we absolutely should have been taught in school, it's how to craft the perfect dating app profile. I mean, if I can still remember the quadratic equation, surely I should have been schooled in the art of snagging a date. But alas, our education system seems to think finding our soulmates isn't as essential as algebra.

2024 April 8, Lydia Wang, Morgan Sullivan, Rachel Varina, “24 Best Hinge Prompts to Actually Help Your Matches Get to Know You”, in Cosmopolitan:

(algebra) A polynomial equation of the second degree; an equation that can be rearranged in standard form as ax²+bx+c=0, where x is an unknown value, a, b, and c are known numbers, and where a does not equal zero.


injective

This term was introduced by Nicolas Bourbaki in his treatise Éléments de mathématique.

short exact sequence
short exact sequence

(algebra, module theory, of a (left) module M over a ring R) Loosely, having a certain generalizing property, abstracted from the study of ℚ as a ℤ-module. Formally, such that any short exact sequence of (left) R-modules beginning with M splits, or any of several equivalent statements: See Injective module.


-adic

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Back-formation from monadic, etc., from Ancient Greek -άς (-ás) (genitive -άδος (-ádos)) + -ικός (-ikós) (English -ad + -ic). Compare related adicity and Latinate -ary.
The algebraic sense is by analogy with p-adic, since ℤ₍ₚ₎ equipped with the (p)-adic topology passes to the ring of p-adic numbers under the operation of completion.

(algebra, of a topology on a module M over a ring R, combined with a symbol (here I but also often p, m, or a) representing an ideal of R) Such that the sets x+Iⁿᴹ (where x∈M, n a non-negative integer) form a basis for the topology.


Boolean algebra

Boolean algebra (plural Boolean algebras)

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

The set of divisors of 30, with binary operators: g.c.d. and l.c.m., unary operator: division into 30, and identity elements: 1 and 30, forms a Boolean algebra.

A Boolean algebra is a De Morgan algebra which also satisfies the law of excluded middle and the law of noncontradiction.

(algebra) An algebraic structure (Σ,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is a unary involutory operator (called "complement"), and 0 and 1 are nullary operators (i.e., constants), such that (Σ,∨,0) is a commutative monoid, (Σ,∧,1) is a commutative monoid, ∧ and ∨ distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)

(algebra, logic, computing) Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or "true" and "false") and are subject to operations based on AND, OR and NOT


Boolean function

Boolean function (plural Boolean functions)

(algebra, logic, computing) Any function based on the operations AND, OR and NOT, and whose elements are from the domain of Boolean algebra

fonction booléenne

function based on AND, OR and NOT operations


two-sided

two-sided inverse — two-sided identity

(algebra) Whose salient properties apply when (the specified object) appears on either side of a given binary operator.


normed

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mathematical analysis
mathematical analysis

(algebra, mathematical analysis) Of a mathematical structure, endowed with a norm.


linear equation

linear equation (plural linear equations)

(algebra) A polynomial equation of the first degree (such as x = 2y - 7)


homology

homology (countable and uncountable, plural homologies)

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Henri Poincaré
Homology
MathWorld
singular homology

From Latin homologia, from Ancient Greek ὁμολογία (homología, “agreement, assent”); compare French homologie. By surface analysis, homo- + -logy.
In topology, first used by French polymath Henri Poincaré, in the sense (close to what is now called a bordism) of a relation between manifolds mapped into a reference manifold: that is, the property of such manifolds that they form the boundary of a higher-dimensional manifold inside the reference manifold. Poincaré's version was eventually replaced by the more general singular homology, which is what mathematicians now mean by homology.

homologia

biology, psychology: relationship of having a common evolutionary or developmental origin

Because of their connection with both homology and cohomology, chain complexes are an important topic of study in homological algebra.

The relationship of being homologous; a homologous relationship.

(algebra) Given a chain complex {Gₙ} and its associated set of homomorphisms {Hₙ}, the rule which explains how each Hₙ maps Gₙ into the kernel of Gₙ₊₁.

同调

algebra: rule by which each member of a chain complex maps into the kernel of the next


automorphism

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Since every linear automorphism of V fixes 0 our interest in the transitivity properties of GL(V) is confined to its action on V* = V - {0}. GL(V) is transitive on V* since any two elements of V* may be chosen as the initial members of two ordered bases; it is not in general 2-transitive because there is no linear automorphism taking an independent pair to a dependent pair.

1971, Norman Biggs, Finite Groups of Automorphisms: Course Given at the University of Southampton, Cambridge University Press, page 25:

In mathematics, automorphisms of algebraic structures play an important role. Automorphisms capture the symmetries inherent in the structures and many important results have been proved by analyzing the automorphism group of the structure.

2005, Maninda Agrawal, Nitin Saxena, “Automorpisms of Finite Rings and Applications to Complexity of Problems”, in Volker Diekert, Bruno Durand, editors, STACS 2005: 22nd Annual Symposium on Theoretical Aspects of Computer Science, Springer, LNCS 3404, page 1:

A tame automorphism is a product of elementary automorphisms, and a non-tame automorphism is called wild. The “tame automorphism problem” asks whether any automorphism is tame.

2014, Alexei Belov, Leonid Bokut, Louis Rowen, Jie-Tai Yu, “The Jacobian Conjecture, Together with Specht and Burnside-Type Problems”, in Ivan Cheltsov, Ciro Ciliberto, Hubert Flenner, James McKernan, Yuri G. Prokhorov, Mikhail Zaidenberg, editors, Automorphisms in Birational and Affine Geometry, Springer, page 274:

(algebra) An isomorphism of a mathematical object or system of objects onto itself.


Galois theory

en-noun

Galois theory (usually uncountable, plural Galois theories)

/ˈɡælwɑ ˈθi.əɹi/

Standardly constructed calque of French théorie de Galois (which was used as a section heading in 1870, Camille Jordan, Traité des substitutions et des équations algébriques). First appeared in print in 1893, Bulletin of the New York Mathematical Society.
Ultimately, named after French mathematician Evariste Galois (1811-1832), who first developed the theory to explore how the roots of a given polynomial equation relate to each other.

In this chapter we present the Galois theory which may be described as the analysis of field extensions by means of automorphism groups.

1989, G. Karpilovsky, Topics in Field Theory, North-Holland, page 299:

The Galois theory of noncommutative rings is a natural outgrowth of the Galois theory of fields.

1989, Donald S. Passman, Infinite Crossed Products, Academic Press, page 297:

1992, Journal of Contemporary Mathematical Analysis, Volume 27, Allerton Press, page 4,
Though often our results are prompted by the classical or parallel Galois theories, their proofs are completely different and are based on the set-theoretical approach.

Classical Galois theory has developed a number of extensions and ramifications into more specific theories, which combine it with other areas of mathematics or restate its main problems in different situations. Three of the most important are geometric Galois theory, differential Galois theory, and Lie Galois theory, all of which have undergone very rapid development in recent years.

2003, “Introduction”, in Leila Schneps, editor, Galois Groups and Fundamental Groups, Cambridge University Press, page ix:

(algebra, field theory) The branch of mathematics dealing with Galois groups, Galois fields, and polynomial equations. It provides a link between field theory and group theory: it permits certain problems in the former to be reduced to the latter, which in some respects is simpler and better understood.


anticommutativity

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(algebra, physics) The state of being anticommutative.


anticommutative

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(algebra) Of an operator * for which a * b = - (b * a)


linear independence

The linear independence of a set of vectors can be determined by calculating the Gram determinant of those vectors; if their Gram determinant is zero, then they are linearly dependent, and if their Gram determinant is non-zero, then they are linearly independent. Incidentally, the same Gram determinant can be used to calculate the hyper-volume of a hyper-parallelepiped (whose edges which "radiate" from an "origin" vertex are described by the vectors).

linearly independent
linearly independent

(algebra) the state of being linearly independent


linearly dependent

(algebra, of a set of vectors or ring elements) Having a nontrivial linear combination which is zero.


linear dependence

linear dependence (plural linear dependences)

(algebra, uncountable) The state of being linearly dependent


symmetric polynomial

symmetric polynomial (plural symmetric polynomials)

x+y+z

xyz

2x²+2y²+2z², etc.

An example of a non-symmetric polynomial is x+2y: if we interchange x and y, this polynomial changes to 2x+y.

(algebra) A polynomial which remains unchanged if its variables are permuted


elementary symmetric polynomial

elementary symmetric polynomial (plural elementary symmetric polynomials)

elementary symmetric polynomials (plural)

(algebra) a coefficient of some power of t in (t+x_1)(t+x_2)⋯(t+x_n), for some n∈ N


subring

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(algebra) A ring which is contained in a larger ring, such that the multiplication and addition on the former are a restriction of those on the latter. When rings are required to have a multiplicative identity (as is standard in most branches of algebra), the subring must also have the same multiplicative identity as the larger ring.


subfield

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Let us describe in general the subfield generated by a given element. Let K be a given field, F a subfield of K, and c an element of K. Consider those elements of K which are given by polynomial expressions of the form
(1)#92;qquadf(c)#61;a#95;0#43;a#95;1c#43;a#95;2c²#43;...#43;a#95;ncⁿ#92;qquad#92;qquad#92;mbox#123;(each#125;a#95;i#92;mbox#123;in#125;F#92;mbox#123;).#125;
[...]
 If f(c) and g(c) ≠ 0 are polynomial expressions like (1), their quotient f(c)/g(c) is an element of K, called a rational expression in c with coefficients in F. The set of all such quotients is a subfield; it is the field generated by F and c and is conventionally denoted by F(c), with round brackets.

1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.1, page 394:

We are now in a position to describe the subfield of K generated by F and our algebraic element u. This subfield F(u) clearly contains the subdomain F[u] of all elements expressible as polynomials f(u) with coefficients in F (cf. (1)). Actually, this domain F[u] is a subfield of K. Indeed, let us find an inverse for any element f(u) ≠ 0 in F[u]. [...] This shows that F[u] is a subfield of K.
 Since, conversely, every subfield of K which contains F and u evidently contains every polynomial f(u) in F[u], we see that F[u] is the subfield of K generated by F and u.

1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.2, page 397:

(algebra) A subring of a field, containing the multiplicative identity and closed under inversion.


homomorphism

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A field homomorphism is a map from one field to another one which is additive, multiplicative, zero-preserving, and unit-preserving.

1954, Kuo-Tsai Chen, Iterated Integrals and Exponential Homomorphisms, Proceedings of the London Mathematical Society, Reprinted in 2001, Philippe Tondeur (editor), Collected Papers of K.-T. Chen, Birkhäuser, page 54,
This motivates a generalization, and exponential homomorphisms are now defined, in an algebraic fashion, from certain free products to formal power series rings with non-commutative indeterminates.

A homomorphism of presheaves h#58;A#92;rightarrowB is a collection of homomorphisms h#95;U#58;A(U)#92;rightarrowB(U) commuting with restrictions.

1997, Glen E. Bredon, Sheaf Theory, 2nd edition, Springer, page 8:

Definition 1.15. Let G and H be matrix Lie groups. A map #92;Phi from G to H is called a Lie group homomorphism if (1) #92;Phi is a group homomorphism and (2) #92;Phi is continuous.

2003, Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, page 17:

(algebra) A structure-preserving map between two algebraic structures of the same type, such as groups, rings, or vector spaces.

automorphism

structure-preserving map between algebraic structures


noetherian

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(algebra) Satisfying some (usually ascending) chain condition:

(of a ring) In which any ascending chain of ideals eventually becomes constant.

(of a module) In which any ascending chain of submodules eventually becomes constant.

(of a group) In which any ascending chain of subgroups eventually becomes constant.

(of a topological space) In which any ascending chain of open subsets eventually becomes constant.


Artinian

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From Artin + -ian, after Emil Artin, an Armenian-Austrian mathematician.

(algebra, of a ring) in which any descending chain of ideals eventually starts repeating.

(algebra, of a module) in which any descending chain of submodules eventually starts repeating.


submodule

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(algebra) A module contained in a larger module, both over the same ring, such that the ring multiplication in the former is a restriction of that in the latter.


local ring

The unique maximal ideal of a (commutative) local ring contains all of the zero divisors of such ring, and all elements of the ring outside of it are units. Then in a local ring, the sum of any two zero divisors is also a zero divisor. Contrapositively, if two ring elements add up to a unit then one of them must be a unit as well. A simple example of a local ring is #92;mathbb#123;Z#125;#95;8.

(algebra) A commutative ring with a unique maximal ideal, or a noncommutative ring with a unique maximal left ideal or (equivalently) a unique maximal right ideal.


nilradical

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(algebra) The set of nilpotent elements of an algebraic structure such as an ideal