chain

From Middle English cheyne, chaine, from Old French chaine, chaene (“chain”), from Latin catēna (“chain”), from Proto-Indo-European *kat- (“to braid, twist; hut, shed”). Doublet of catena.

der

a chain is only as strong as its weakest link

We first find an approximation of the chain partition, i.e. a small but not minimum size set of chains which cover all elements of the poset.

2003, Jeremy P. Spinrad, Efficient Graph Representations, American Mathematical Society, page 108:

(mathematics, set theory, order theory) A totally ordered set, especially a totally ordered subset of a poset.


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.

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).


ring

(mathematics, order theory) A family of sets closed under finite union and finite intersection.


order

en-noun

order (countable and uncountable, plural orders)

→ 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

(order theory) The relation with which a partially ordered set is equipped.


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).

homogenous

may be considered incorrect; see usage note at §§ Usage notes

(set theory, order theory, of a relation) Holding between a set and itself; being an endorelation.


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, order theory) A partially ordered set in which every pair of elements has a unique supremum and a unique infimum.


atomic

(order theory, of a partially ordered set with a least element 0) Such that for every element b>0 there exists an atom a such that b≥a>0.


filter

→ Japanese: フィルター (firutā)

→ Korean: 필터 (pilteo)

From Middle English filtre, from Medieval Latin filtrum (compare also Old French feutre (“felt; filter”)), from Frankish *filtir, from Proto-West Germanic *felt. See felt. Doublet of phin.

The collection of cofinite subsets of ℝ is a filter under inclusion: it includes the intersection of every pair of its members, and includes every superset of every cofinite set.

If (1) the universal set (here, the set of natural numbers) were called a "large" set, (2) the superset of any "large" set were also a "large" set, and (3) the intersection of a pair of "large" sets were also a "large" set, then the set of all "large" sets would form a filter.

(mathematics, order theory) A non-empty upper set (of a partially ordered set) which is closed under binary infima (a.k.a. meets).


strict

en-adj

strict (comparative stricter or more strict, superlative strictest or most strict)

stricter (comparative)
more strict (comparative)
strictest (superlative)
most strict (superlative)

Borrowed from Latin strictus, past participle of stringere (“to draw tight, bind, contract”). Doublet of strait and stretto. See stringent, strain.

(set theory, order theory) Irreflexive; if the described object is defined to be reflexive, that condition is overridden and replaced with irreflexive.


binary relation

en-noun

binary relation (plural binary relations)

A partially ordered set #x5C;langleA,#x5C;varrho#x5C;rangle consists of a nonvoid set A and a binary relation #x5C;varrho on A, such that #x5C;varrho satisfies properties (P1)-(P3).

1978, George Grätzer, General Lattice Theory, Academic Press, page 1:

1.30. Corollary. If P is a binary relation which is asymmetric and negatively transitive, then P is also transitive.
It should be noted that a binary relation may be irreflexive and negatively transitive without being transitive; as an example, consider the standard inequality relation (≠).

1999, James C. Moore, Mathematical Methods for Economic Theory 1, Springer, page 24:

Definition If E is a non-empty set then by an order on E we mean a binary relation on E that is reflexive, anti-symmetric, and transitive.

2005, T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, page 1:

(set theory, order theory, "on" a set A) A subset of the Cartesian product A×A (the set of ordered pairs (a, b) of elements of A).

(set theory, order theory, "on" or "between" sets A and B) A subset of the Cartesian product A×B.


bounded

en-adj

bounded (comparative more bounded, superlative most bounded)

more bounded (comparative)
most bounded (superlative)

(set theory, order theory, of a poset X with partial order ≤) That contains a least element, a, and a greatest element, b, such that for all x ∈ X, a ≤ x ≤ b.

barita

that contains a least and a greatest elements


poset

1973, Barbara L. Osofsky, Homological Dimensions of Modules, American Mathematical Society, →ISBN, page 76,
42. Definition. A poset (partially ordered set) (X, ≤) (usually written just X) is a set X together with a transitive, antisymmetric relation ≤ on X.
43. Definition. A linearly ordered set or chain is a poset (X, ≤), such that ∀a, b ∈ X, either a ≤ b or b ≤ a or a = b.

1998, Yuri A. Drozd, Representations of bisected posets and reflection functors, Idun Reiten, Sverre O. Smalø, Øyvind Solberg (editors), Algebras and Modules II, American Mathematical Society (for Canadian Mathematical Society), page 153,
We construct a complete set of reflection functors for the representations of posets and prove that they really have the usual properties. In particular, when the poset is of finite representation type, all of its indecomposable representations can be obtained from some "trivial" ones via relations. To define such reflection functors, a wider class of matrix problem is introduced, called "representations of bisected posets".

The combinatorial interest in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider here) and the covering graph.

1999, Manfred Stern, Semimodular Lattices: Theory and Applications, Cambridge University Press, page 189:

(set theory, order theory) A partially ordered set.


partial order

1986, Kenneth R. Goodearl, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, Softcover reprint 2010, page xxi,
A partial order on a set X is any reflexive, antisymmetric, transitive relation on X. In most cases, partial orders are denoted ≤.

1999, Paul A. S. Ward, An Online Algorithm for Dimension-Bound Analysis, Patrick Amestoy, P. Berger, M. Daydé, I. Duff, V. Frayssé, L. Giraud, D. Ruiz (editors), Euro-Par ’99 Parallel Processing: 5th International Euro-Par Conference, Proceedings, Springer, LNCS 1685, page 144,
The vector-clock size necessary to characterize causality in a distributed computation is bounded by the dimension of the partial order induced by that computation.

Consider an arbitrary finite set S. The family #x5C;mathcal#x7B;P#x7D; of all strict partial orders (asymmetric, transitive, cf. 1.8.3, p. 14) on S enjoys a remarkable property: any partial order P can be linked to any other partial order P’ by a sequence of steps each of which consists of changing the order either by adding one ordered pair of elements of S (imposing an ordering between two previously-incomparable elements) or by removing one ordered pair (causing two previously related elements to become incomparable), without ever leaving the family #x5C;mathcal#x7B;P#x7D;.

2008, David Eppstein, Jean-Claude Falmagne, Sergei Ovchinnikov, Media Theory: Interdisciplinary Applied Mathematics, Springer, page 7:

(set theory, order theory) (informal) An ordering of the elements of a collection that behaves like that of the natural numbers by size, except that some elements may not be comparable (if all elements are comparable, it is called a total order); (formal) a binary relation that is reflexive, antisymmetric, and transitive.


partially ordered

This leads to the introduction of new kinds of abstract spaces—partially ordered linear spaces—and to their systematic use in functional analysis. The beginnings of a theory of partially ordered linear spaces are given in the works of L. V. Kantorovič in 1935-1937.

1963, L. V. Kantorovič, B. Z. Vulih, A. G. Pinsker, “Partially Ordered Groups and Partially Ordered Linear Spaces”, in A. L. Brudno, editor, American Mathematical Society Translations, Series 2, Volume 27: 18 papers on algebra, American Mathematical Society, page 51:

The chains in #x5C;mathcal#x7B;V#x7D; form a class #x5C;mathcal#x7B;C#x7D; which is partially ordered by inclusion.

1966, S. J. Taylor, Introduction to Measure and Integration, Cambridge University Press, page 22:

2008, Patrik Eklund, M. Ángeles Galán, Partially Ordered Monads and Rough Sets, James F. Peters, Andrzej Skowron (editors), Transactions on Rough Sets VIII, Volume 8, Springer, LNCS 5084, page 53,
In this paper we will show that partially ordered monads contain appropriate structure for modeling rough sets in a generalized relational setting.

(set theory, order theory, of a set) Equipped with a partial order; when the partial order is specified, often construed with by.


partially ordered set

en-noun

partially ordered set (plural partially ordered sets)

半順序集合

はんじゅんじょしゅうごう

(set theory, order theory, loosely) A set that has a given, elsewhere specified partial order.

1959 [D. Van Nostrand], Edward James McShane, Truman Arthur Botts, Real Analysis, 2005, Dover, page 28,
A partially ordered set means a pair (P,≻) consisting of a set P and a partial order ≻ in P. As usual, when the meaning is clear, we may suppress the notation of "≻" and speak of the partially ordered set P.
The ordered fields defined earlier are easily seen to be examples of partially ordered sets.

In sections 7-10 we shall consider random fields over some subsets T of the partially ordered set T_M.

1994, I. V. Evstigneev, P. E. Greenwood, Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting, American Mathematical Society, page 35:

The invention of a derivative of a finite partially ordered set by Nazarova and Roiter in the late 1960s or early 1970s was a seminal event in the subject of representations of finite partially ordered sets (see [Simson 92]).

2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 45:

(set theory, order theory, formally) The ordered pair comprising a set and its partial order.


antisymmetric

1987, David C. Buchthal, Douglas E. Cameron, Modern Abstract Algebra, Prindle, Weber & Schmidt, page 479,
The standard example for an antisymmetric relation is the relation less than or equal to on the real number system.

(i) The identity relation on a set A is an antisymmetric relation.
(ii) Let R be a relation on the set N of natural numbers defined by
 x R y #x5C;Leftrightarrow 'x divides y' for all x, y ∈ N.
This relation is an antisymmetric relation on N.

2006, S. C. Sharma, Metric Space, Discovery Publishing House, page 73:

(set theory, order theory, of a binary relation R on a set S) Having the property that, for any two distinct elements of S, at least one is not related to the other via R; equivalently, having the property that, for any x, y ∈ S, if both xRy and yRx then x=y.

antisymetrický

(order theory; of a binary relation on a set)


total order

[…]we conclude §2.1 by showing how, given a triangulation #x5C;Delta (i.e., simplicial decomposition) of a closed oriented 3-manifilld M, and a total order '#x5C;le' on the set of vertices of #x5C;Delta, as well as a choice of a system #x5C;mathcal#x7B;B#x7D; of orthonormal bases for various Hilbert spaces that get specified in the process, we may obtain a complex number #x5C;langle(M,#x5C;Delta,#x5C;le,#x5C;mathcal#x7B;B#x7D;)#x5C;rangle.

2001, Vijay Kodiyalam, V. S. Sunder, Topological Quantum Field Theories from Subfactors, CRC Press (Chapman & Hall), page 2:

Example 6.2.2. Suppose A is a finite set and R is a partial order on A. Prove that R can be extended to a total order on A. In other words, prove that there is a total order T on A such that R ⊆ T.

2006, Daniel J. Velleman, How to Prove It: A Structured Approach, 2nd edition, Cambridge University Press, page 269:

2013, Nick Huggett, Tiziana Vistarini, Christian Wüthrich, 15: Time in Quantum Gravity, Adrian Bardon, Heather Dyke (editors), A Companion to the Philosophy of Time, Wiley, 2016, Paperback, page 245,
A binary relation R defines a total order on a set X just in case for all x, y, z ∈ X, the following four conditions obtain: (1) Rxx (reflexivity), (2) Rxy & Ryz → Rxz (transitivity), (3) Rxy & Ryx → x = y (weak antisymmetry), and (4) Rxy ∨ Ryx (comparability). Bearing in mind that the relata of the total order are not events in ℰ, but entire equivalence classes ℰ/S of simultaneous events, it is straightforward to ask ≤ to be a total order of ℰ/S.

(set theory, order theory) A partial order, ≤, (a binary relation that is reflexive, antisymmetric, and transitive) on some set S, such that any two elements of S are comparable (for any x, y ∈ S, either x ≤ y or y ≤ x).

lineární uspořádání

partial order that applies an order to any two elements

linear order

partial order which applies an order to any two elements


monotone function

monotone function (plural monotone functions)

monotone Funktion

order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)

mathematical analysis
mathematical analysis

(order theory, mathematical analysis) A function f : X→Y (where X and Y are posets with partial order "≤") with either: (1) the property that x ≤ y implies f(x) ≤ f(y), or (2) the property that x ≤ y implies f(y) ≤ f(x).


preorder

The relation of logical implication over sentences is an example of a preorder.

Example 436 For each set A, the relation #x5C;subseteq is a preorder on #x5C;mathcal#x7B;P#x7D;(A).

2002, Yves Nievergelt, Foundations of Logic and Mathematics, Springer (Birkhäuser), page 152:

Let us first compare S#x5F;1, S#x5F;2 and S#x5F;3 w.r.t. the partial preorder #x5C;succeq.

2010, S. Kaci, “Refined Preference-Based Argumentation Frameworks”, in Pietro Baroni, F. Cerutti, M. Giacomin, G. R. Simari, editors, Computational Models of Argument: Proceedings of COMMA 2010, IOS Press, page 306:

2000, Jean-Charles Pomerol, Sergio Barba-Romero, Multicriterion Decision in Management: Principles and Practice, Springer, Softcover, page 58,
It can easily be verified that the above relation is a preorder, i.e. that it is reflexive and transitive.

(set theory, order theory) A binary relation that is reflexive and transitive.


well-order

1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237,
̲X is well-order enriched iff every morphism set ̲X(X,Y) carries a well-order ≤_(XY) such that
f≨_(XY)g⇒h•f≨_(XY)h•g
for every h:Y→Z.

2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71,
Some simple facts and terminology about well-orders were already given in and just before 1.8.4. Here are some more: In a well-order A, every element x is clearly of just one of these three kinds: x is the first element; x is a successor element - i.e., x has an immediate predecessor; or x is a limit element - i.e., x has a predecessor but no immediate predecessor. The structure (∅, ∅) is a well-order.

2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), page 378,
Definition 1226 (Von Neumann Well-Orders). A well-order X is said to be a von Neumann well-order if for every x∈X, we have x=y∈X|y<x (that is x is equal to the set Pred(x) consisting of its predecessors).
Clearly the examples listed by von Neumann above, namely
empty , empty, empty ,empty, empty ,empty,empty ,empty, …
are all von Neumann well-orders if ordered by the membership relation "∈," and the process can be iterated through the transfinite. Our immediate goal is to show that these and only these are the von Neumann well-orders, with exactly one von Neumann well-order for each ordinal (order type of a well-order). This is called the existence and uniqueness result for the von Neumann well-orders.

(set theory, order theory) A total order of some set such that every nonempty subset contains a least element.


well-order

en-verb

well-order (third-person singular simple present well-orders, present participle well-ordering, simple past and past participle well-ordered)

well-orders (present singular third-person)
well-ordering (participle present)
well-ordered (participle past)
well-ordered (past)

The set of positive integers is well-ordered by the relation ≤.

1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, 2006, Dover (Dover Phoenix), page 111,
Starting from these special well-ordered subsets, it is then possible to well-order the entire set.

1975 [The Williams & Wilkins Company], Dennis Sentilles, A Bridge to Advanced Mathematics, Dover, 2011, page 182,
To carry the analogy a bit further, the axiom of choice implies the ability to well order any set.

Then #x5C;le#x5F;C is a well defined order on C, and (C,#x5C;le#x5F;C) belongs to #x5C;mathcal#x7B;X#x7D; (that is, #x5C;le#x5F;C well orders C) and is an upper bound for #x5C;mathcal#x7B;C#x7D;.

2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, page 18:

(set theory, order theory, transitive) To impose a well-order on (a set).


antichain

First of all, Zorn's Lemma implies that every uncountable antichain in T is contained in a maximal uncountable antichain. So it suffices to make sure that T has no maximal uncountable antichains.

1997, Winfried Just, Martin Weese, Discovering Modern Set Theory II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 140:

2013, Vijay K. Garg, Maximal Antichain Lattice Algorithms for Distributed Computations, Proceedings, Davide Frey, Michel Raynal, Saswati Sarkar, Rudrapatna K. Shyamasundar, Prasun Sinha (editors), Distributed Computing and Networking: 14th International Conference, ICDCN, Springer, LNCS 7730, page 245,
We first define three different but isomorphic lattices: the lattice of maximal antichain ideals, the lattice of maximal antichains and the lattice of strict ideals.

2014, Martin Aigner, Günter M. Ziegler, Proofs from THE BOOK, Springer, 5th Edition, page 199,
In 1928 Emanuel Sperner asked and answered the following question: Suppose we are given the set N=1,2,3,…,n. Call a family ℱ of subsets of N, which has partial order ⊆]) an antichain if no set of ℱ contains another set of the family ℱ. What is the size of a largest antichain? Clearly, the family ℱₖ of all k-sets satisfies the antichain property with |ℱₖ|=\binomnk. Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size \binomn[n/2]= max ₖ\binomnk. Sperner's theorem now asserts that there are no larger ones.

(set theory, order theory, graph theory) A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤.


preordered

Most of the theory of this section on linear order spaces (and cones) can be generalized to preordered linear spaces (and wedges).

1980, Johann Schröder, Operator Inequalities, Academic Press, page 8:

It can be proved [see Herden (1989a,c, 1990, 1993a,b) and Herden and Mehta (1997a) that a continuous isotone (or strictly isotone) function defined on a closed subset K of a preordered topological space X has a continuous isotone (or strictly isotone) extension to a function g on X if and only if there is a separable system on X which satisfies certain "separation" conditions.

1998, Ghanshyam B. Mehta, “Chapter 1: Preference and Utility”, in Salvador Barbera, Peter Hammond, Christian Seidl, editors, Handbook of Utility Theory, Volume 1: Principles, Springer, page 30:

We remind the reader that the barycentric subdivision sd(P) of a preordered set P is the poset of chains in P, ordered by inclusion.

1999, Rafael Villarroel-Flores, Equivariant Homotopy Type of Categories and Preordered Sets, University of Minnesota, page 24:

(set theory, order theory, of a set) Equipped with a preorder.


ultrafilter

An ultrafilter is maximal in the sense that if any other element of the poset not already in it were added to it, one could deduce (from the laws which define the filter, and the given ordering relation, i.e., the structure of the poset) that the resulting filter must be improper; i.e., it must contain all the elements of the poset.

(mathematics, order theory) A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter.


cofinal

The Archimedean property of real numbers means that the natural numbers form a cofinal subset of ℝ.

In order to apply this result to Quine’s project of radical translation, we take X to be the set of all Native observation sentences and construe an observational sub-language that contains only observation sentences and all their syntactic parts such that X is cofinal in the set of expressions of the sub-language.

2007 November 29, Manfred Kupffer, “An Unintentional Defense of the Indeterminacy of Meaning?”, in Erkenntnis, volume 68, number 2, →DOI:

partially ordered set
partially ordered set#English

(order theory) Of a subset of a partially ordered set; containing elements at least as late as any given element of the set, relative to the given partial order.

współkońcowy

containing elements at least as late as any given element of the containing set


Heyting algebra

en-noun

Heyting algebra (plural Heyting algebras)

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

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

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

which is just obvious.

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

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

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

Arend Heyting
Heyting algebra
intuitionistic logic

algebra di Heyting

bounded lattice equipped with operation called implies


upper set

(mathematics, order theory) A subset of a poset which contains any ascending chain which starts at any element of itself.


lower set

(mathematics, order theory) A subset of a poset which contains any descending chain which starts at any element of itself.


De Morgan algebra

De Morgan algebra (plural De Morgan algebras)

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

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

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

By a topological de Morgan algebra we shall mean an abstract algebra (A,#x5C;land,#x5C;lor,#x5C;',l) where (A,#x5C;land,#x5C;lor,l) is a de Morgan algebra,

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

If (L,#x5C;le,#x7B;⁻#x7D;) is a bounded distributive lattice with negation function (resp. a De Morgan algebra), then (L#x5C;langle#x5C;#x21;#x5C;langleS#x5C;rangle#x5C;#x21;#x5C;rangle,#x5C;le,#x7B;⁻#x7D;) constitutes also a bounded distributive lattice with negation function (resp. a De Morgan algebra); for every r#x5C;inL#x5C;langle#x5C;#x21;#x5C;langleS#x5C;rangle#x5C;#x21;#x5C;rangle its negation #x5C;overline#x7B;r#x7D;#x5C;inL#x5C;langle#x5C;#x21;#x5C;langleS#x5C;rangle#x5C;#x21;#x5C;rangle is defined by (#x5C;overline#x7B;r#x7D;,s)#x3D;#x5C;overline#x7B;(r,s)#x7D; for every s#x5C;inS.

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

distributive lattice
distributive lattice

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

Augustus De Morgan
De Morgan algebra
Grigore Moisil


well-orderable

(set theory, order theory) Capable of being well-ordered.


linear order

A set of numbers on a number line form a linear order.


Galois connection

Galois connection (plural Galois connections)

From Galois (attributive form of Galois theory) + connection; ultimately after French mathematician Évariste Galois. Coined by Norwegian mathematician Øystein Ore in 1944, Galois connexions, Transactions of the American Mathematical Society, 55, pages 493-513.

1986, Horst Herrlich, Miroslav Hušek, Galois Connections, Austin Melton, Mathematical Foundation of Programming Semantics: International Conference, Proceedings, Springer, Lecture Notes in Computer Science: 239, page 122,
Define maps G:A→B and F:B→A by G(a)=y!∈!Y|∀x!∈!axρy and F(b)=x!∈!X|∀y!∈!bxρy. Then (F,G) is called a Galois connection of the first kind.

Finally, we discuss Galois connections. It is interesting to note that these definitions can be found in Birkhoff (1940, 1948, 1967) under the heading of "polarities," and Everett (1944) showed that all Galois connections defined on power sets can be obtained from polarities.

2001, J. Michael Dunn, Gary M. Hardegree, Algebraic Methods in Philosophical Logic, Oxford University Press, page 398:

Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair #x5C;left#x5C;langle#x5C;uparrow,#x5C;downarrow#x5C;right#x5C;rangle of mappings #x5C;uparrow assigning subcollections of Y to subcollections of X, and #x5C;downarrow assigning subcollections of X to subcollections of Y.

2006, Radim Bělohlávek, Taťána Funioková, Vilém Vychodil, “Galois connections with Truth Stressers: Foundations for Formal Concept Analysis of Object-Attribute Data with Fuzzy Attributes”, in Bernd Reusch, editor, Computational Intelligence, Theory and Applications: International Conference, Proceedings, Springer,, page 205:

(category theory, order theory) A type of correspondence between partially ordered sets (posets), also applicable to preordered sets.


transitive closure

transitive closure (plural transitive closures)

Transitive closure
Transitive set#Transitive closure

(set theory, order theory, of a binary relation R on a set X) The smallest binary relation on X that includes R and is transitive.


totally ordered

1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67,
(A.2.5) THEOREM If A is a totally ordered ring and if I is a proper order ideal, then A/I is a totally ordered ring (with the operations and order given above).

The set {1, 2, 3, 4, 6, 8, 12, 24}, ordered by 'divides' is not totally ordered, since (for example) 6 and 8 are not related. However, the set {1, 2, 4, 12, 24} is totally ordered by the relation 'divides'.

1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91:

G is a transitive group of order automorphisms of a totally ordered set X,
θ is a convex G-congruence on the totally ordered set X,
ξ is the order type of the totally ordered set X,
◌̅η is the order type of some class Y=xη of the congruence η,

1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation

ζ is the order type of the totally ordered quotient set X/θ of the totally ordered set X by the congruence θ.

(set theory, order theory) That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset.


bounded lattice

bounded lattice (plural bounded lattices)

2004, Anna Maria Radzikowska, Etienne Kerre, On L-Fuzzy Rough Sets, Leszek Rutkowski, Jörg Siekmann, Ryszard Tadeusiewicz, Lotfi A. Zadeh (editors), Artificial Intelligence and Soft Computing — ICAISC 2004: 7th International Conference, Proceedings, Springer, LNAI 3070, page 526,
A residuated lattice is an extension of a bounded lattice by a monoid operation and its residuum, which are abstract counterparts of a triangular norm and a fuzzy residual implication, respectively.

2006, Bart Van Gasse, Chris Cornelis, Glad Deschrijver, Etienne Kerre, Triangle Lattices: Towards an Axiomatization of Interval-Valued Residuated Lattices, Salavatore Greco, Yukata Hata, Shoji Hirano, Masahiro Inuiguchi, Sadaaki Miyamoto, Hung Son Nguyen, Roman Słowiński (editors), Rough Sets and Current Trends in Computing: 5th International Conference, Proceedings, Springer, LNAI 4259, page 117,
Indeed, in the scope of these logics, formulas can be assigned not only 0 and 1 as truth values, but also elements of [0,1], or, more generally, of a bounded lattice ℒ.

In this paper, we study idempotent nullnorms on bounded lattices. We prove that there is no idempotent nullnorm on a distributive bounded lattice L different from the proposal in [13].

2018, Gül Deniz Çayli, Funda Karaçal, “Some Remarks on Idempotent Nullnorms on Bounded Lattices”, in Vicenç Torra, Radko Mesiar, Bernard De Baets, editors, Aggregation Functions in Theory and in Practice, Springer,, page 32:

(algebra, order theory) Any lattice (type of partially ordered set) that has both a greatest and a least element.


ordered ring

1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217,
If < is an ordering on A compatible with its ring structure, we shall say that (A,+,·,<) is an ordered ring. An element x of an ordered ring A is positive if x>0, and x is strictly positive if x>0.
The set of all positive elements of an ordered ring A is denoted by A_+, and the set of all strictly positive elements of A is denoted by A^*₊.
If (A,+,·,<) is an ordered ring and if < is a total ordering, we shall, of course, call (A,+,·,<) a totally ordered ring; if (A,+,·) is a field, we shall call (A,+,·,<) an ordered field, and if, moreover, < is a total ordering, we shal call (A,+·,<) a totally ordered field.

(OR) The relations x>0 and y>0 imply xy>0.
The ring A, together with such an ordering, is called an ordered ring.
Examples. — 1) The rings Q and Z , with the usual orderings, are ordered rings.
2) A product of ordered rings, equipped with the product ordering, is an ordered ring. In particular, the ring Aᴱ of mappings from a set E to an ordered ring A is an ordered ring.
3) A subring of an ordered ring, with the induced ordering, is an ordered ring.

1990, P. M. Cohn, J. Howie (translators), Nicolas Bourbaki, Algebra II: Chapters 4-7, [1981, N. Bourbaki, Algèbre, Chapitres 4 à 7, Masson], Springer, 2003, Softcover reprint, page 19, DEFINITION 1. — Given a commutative ring A, we say that an ordering on A is compatible with the ring structure on A if it is compatible with the additive group structure of A, and if it satisfies the following axiom

(algebra, order theory, ring theory) A ring, R, equipped with a partial order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.

The positive elements in an ordered ring allow us to compare elements to 0, but we know in the integers that we can compare any two elements to each other. For example, we know that 4#x3E;2 because 4-2#x3E;0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b#x5C;inR, then we know by trichotomy that exactly one of the following must be true: a-b#x3E;0, a-b#x3D;0, or -(a-b)#x3E;0.

2013, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom, Abstract Algebra: An Inquiry Based Approach, CRC Press, page 253:

(1) The set R⁺ is closed under addition and multiplication.
(2) If x∈R then exactly one of the following is true: (trichotomy law)
(a) x=0,
(b) x∈R⁺,
(c) -x∈R⁺.
If further R is an integral domain we call R an ordered integral domain.
[…]
Lemma 3.5.9. If R is an ordered ring and a∈R is a positive element, then the set na:n∈ N⊂R⁺.
[…]
Theorem 3.5.2. An ordered ring must be infinite.

2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra, Johns Hopkins University Press, page 77, Definition 3.5.4. A ring R is an ordered ring if there exists a distinguished set R⁺, R⁺⊂R, called the set of positive elements, with the properties that

(algebra, order theory, ring theory) A ring, R, equipped with a total order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.


algebraic poset

algebraic poset (plural algebraic posets)

1985 October, Rudolf-E. Hoffmann, The Injective Hull and the 𝒞ℒ-Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,
A poset (P,<) is said to be algebraic if and only if
i) P is up-complete, i.e., for every non-empty up-directed subset D,thesupremum operatorname supd exists,
ii) for every x∈P, the set
K_X:=y∈P|ycompact,y<x
is non-empty and up-directed, and
x= operatorname supK_x.
A poset P is an algebraic poset if and only if it is a continuous poset in which, for every x,y∈P,x≪y (if and) only if x<c<y for some compact element c of P.
Concerning the definition of an algebraic poset, a caveat may be in order (which, mutatis mutandis, applies to continuous posets): it may happen that all of the axioms for an algebraic poset are satisfied except that the sets K_x fail to be up-directed ([50], 4.2 or [49], 4.5). Even when "enough" compact sets are readily available, it sometimes remains a delicate problem to verify the up-directedness of the sets K_x.
The concept of an algebraic poset arose in theoretical computer science ([50], [54], cf. also [14]). It is a natural extension of he familiar notion of a (complete) "algebraic lattice" (cf. [9], [20], I-4).

ALG is the category whose objects are algebraic posets and whose morphisms are continuous functions.
A structure (D,#x5C;le#x5F;D,#x5C;cup#x5F;D) where (D,#x5C;le#x5F;D) is an algebraic poset and #x5C;cup#x5F;D is a binary operation on D which is continuous (in both variables), commutative, associative and absorptive (i.e., d#x5C;cup#x5F;Dd#x3D;d for all d#x5C;inD) will be called a nondeterministic algebraic poset.

1988, Karel Hrbacek, “A Powerdomain Construction”, in Michael Main, Austin Melton, Michael Mislove, David Schmidt, editors, Mathematical Foundations of Programming Language Semantics: 3rd Workshop, Proceedings, Springer, page 202:

It is important to note that we have not assumed an algebraic poset is a cpo. The canonical example of an algebraic poset is #x7B;A#x2A;#x5C;cupA#x2A;#x7D;#x5C;sqrt#x7B;#x7D; in the prefix order; this poset enjoys the added condition that every element is compact.

1992, Stephen D. Brookes et al., editors, Mathematical Foundations of Programming Semantics: 7th International Conference, Proceedings, Springer, page 88:

The correspondence between primes and occurrences suggests that given an abstract prime algebraic poset, we may construct a behavioural presentation from it.

1997, Michael W. Shields, Semantics of Parallelism: Non-Interleaving Representation of Behaviour, Springer, page 41:

(algebra, order theory) A partially ordered set (poset) in which every element is the supremum of the compact elements below it.


Hausdorff gap

Named after German mathematician Felix Hausdorff (1868–1942), who published proof of the first example in 1909.

The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.

Of course, Hausdorff did not talk of models of set theory or prove absoluteness results. He gave a direct construction in ZFC of what is now called a Hausdorff gap and proved properties 1 through 4 for his construction. The notion of a Hausdorff gap and the proof that Hausdorff gaps are indestructible under cardinal preserving extensions (property 5 above) are due to Kunen (unpublished).

1995, Robert M. Solovay, “*1970 [Godel's ontological proof]”, in Kurt Gödel, edited by Solomon Feferman, John W. Dawson Jr., Warren Goldfarb, Charles Parsons, and Robert M. Solovay, Kurt Gödel: Collected Works: Volume III, Oxford University Press, page 419:

Let #x5C;langle#x5C;langlea#x5F;#x5C;epsilon#x3A;#x5C;epsilon#x3C;#x5C;omega#x5F;1#x5C;rangle#x5C;langleb#x5F;#x5C;epsilon#x3A;#x5C;epsilon#x3C;#x5C;omega#x5F;1#x5C;rangle#x5C;rangle be a Hausdorff gap.

1997, Winfried Just, Martin Weese, Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 118:

2013, Ilijas Farah, Eric Wofsey, 3: Set theory and operator algebras, James Cummings, Ernest Schimmerling (editors), Appalachian Set Theory: 2006-2012, London Mathematical Society, Cambridge University Press, page 101,
This family is one of the instances of incompactness of ω₁ that are provable in ZFC, along with Hausdorff gaps, special Aronszajn trees, or nontrivial coherent families of partial functions.

(set theory, order theory) A pair of collections of integer sequences such that there is no integer sequence lying between the two.


endorelation

(mathematics, order theory) A binary relation between a set and itself.