Look upAppendix:Glossary of order theory in Wiktionary, the free dictionary.

This is a glossary of some terms used in various branches ofmathematics that are related to the fields oforder,lattice, anddomain theory. Note that there is a structuredlist of order topics available as well. Other helpful resources might be the following overview articles:

• completeness properties of partial orders
• distributivity laws of order theory

In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context,${\displaystyle \le }$ will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote thestrict order induced by${\displaystyle \le .}$

• Acyclic. Abinary relation is acyclic if it contains no "cycles": equivalently, itstransitive closure isantisymmetric.^[1]
• Adjoint. SeeGalois connection.
• Alexandrov topology. For a preordered setP, any upper setO isAlexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open.
• Algebraic poset. A poset is algebraic if it has a base of compact elements.
• Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elementsx andy such thatx ≤y. In other words, the order relation of an antichain is just the identity relation.
• Approximates relation. Seeway-below relation.
• Antisymmetric relation. Ahomogeneous relationR on a setX isantisymmetric, ifx R y andy R x impliesx = y, for all elementsx,y inX.
• Antitone. Anantitone functionf between posetsP andQ is a function for which, for all elementsx,y ofP,x ≤y (inP) impliesf(y) ≤f(x) (inQ). Another name for this property isorder-reversing. Inanalysis, in the presence oftotal orders, such functions are often calledmonotonically decreasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is calledmonotone ororder-preserving.
• Asymmetric relation. Ahomogeneous relationR on a setX is asymmetric, ifx R y impliesnot y R x, for all elementsx,y inX.
• Atom. An atom in a posetP with least element 0, is an element that is minimal among all elements that are unequal to 0.
• Atomic. An atomic posetP with least element 0 is one in which, for every non-zero elementx ofP, there is an atoma ofP witha ≤x.

• Base. Seecontinuous poset.
• Binary relation. A binary relation over two sets${\displaystyle X\text{ and }Y}$ is a subset of theirCartesian product${\displaystyle X\times Y.}$
• Boolean algebra. A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every elementx has a complement ¬x, such thatx ∧ ¬x = 0 andx ∨ ¬x = 1.
• Bounded poset. Abounded poset is one that has a least element and a greatest element.
• Bounded complete. A poset isbounded complete if every of its subsets with some upper bound also has a least such upper bound. The dual notion is not common.

• Chain. A chain is a totally ordered set or a totally ordered subset of a poset. See alsototal order.
• Chain complete. Apartially ordered set in which everychain has aleast upper bound.
• Closure operator. A closure operator on the posetP is a functionC :P →P that is monotone,idempotent, and satisfiesC(x) ≥x for allx inP.
• Compact. An elementx of a poset is compact if it isway below itself, i.e.x<<x. One also says that such anx isfinite.
• Comparable. Two elementsx andy of a posetP are comparable if eitherx ≤y ory ≤x.
• Comparability graph. The comparability graph of a poset (P, ≤) is thegraph with vertex setP in which the edges are those pairs of distinct elements ofP that are comparable under ≤ (and, in particular, under its reflexive reduction <).
• Complete Boolean algebra. ABoolean algebra that is a complete lattice.
• Complete Heyting algebra. AHeyting algebra that is a complete lattice is called a complete Heyting algebra. This notion coincides with the conceptsframe andlocale.
• Complete lattice. A completelattice is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.
• Complete partial order. A complete partial order, orcpo, is adirected complete partial order (q.v.) with least element.
• Complete relation. Synonym forConnected relation.
• Complete semilattice. The notion of acomplete semilattice is defined in different ways. As explained in the article oncompleteness (order theory), any poset for which either all suprema or all infima exist is already a complete lattice. Hence the notion of a complete semilattice is sometimes used to coincide with the one of a complete lattice. In other cases, complete (meet-) semilattices are defined to bebounded completecpos, which is arguably the most complete class of posets that are not already complete lattices.
• Completely distributive lattice. A complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets.
• Completion. A completion of a poset is anorder-embedding of the poset in a complete lattice.
• Completion by cuts. Synonym ofDedekind–MacNeille completion.
• Connected relation. A total or complete relationR on a setX has the property that for all elementsx,y ofX, at least one ofx R y ory R x holds.
• Continuous poset. A poset is continuous if it has abase, i.e. a subsetB ofP such that every elementx ofP is the supremum of a directed set contained in {y inB |y<<x}.
• Continuous function. SeeScott-continuous.
• Converse. The converse <° of an order < is that in which x <° y whenever y < x.
• Cover. An elementy of a posetP is said to cover an elementx ofP (and is called a cover ofx) ifx <y and there is no elementz ofP such thatx <z <y.
• cpo. Seecomplete partial order.

• dcpo. Seedirected complete partial order.
• Dedekind–MacNeille completion. The Dedekind–MacNeille completion of apartially ordered set is the smallestcomplete lattice that contains it.
• Dense order. Adense posetP is one in which, for all elementsx andy inP withx <y, there is an elementz inP, such thatx <z <y. A subsetQ ofP isdense inP if for any elementsx <y inP, there is an elementz inQ such thatx <z <y.
• Derangement. A permutation of the elements of a set, such that no element appears in its original position.
• Directed set. Anon-empty subsetX of a posetP is called directed, if, for all elementsx andy ofX, there is an elementz ofX such thatx ≤z andy ≤z. The dual notion is calledfiltered.
• Directed complete partial order. A posetD is said to be a directed complete poset, ordcpo, if every directed subset ofD has a supremum.
• Distributive. A latticeL is called distributive if, for allx,y, andz inL, we find thatx ∧ (y ∨z) = (x ∧y) ∨ (x ∧z). This condition is known to be equivalent to its order dual. A meet-semilattice is distributive if for all elementsa,b andx,a ∧b ≤x implies the existence of elementsa' ≥a andb' ≥b such thata' ∧b' =x. See alsocompletely distributive.
• Domain. Domain is a general term for objects like those that are studied indomain theory. If used, it requires further definition.
• Down-set. Seelower set.
• Dual. For a poset (P, ≤), the dual orderP^d = (P, ≥) is defined by settingx ≥ yif and only ify ≤ x. The dual order ofP is sometimes denoted byP^op, and is also calledopposite orconverse order. Any order theoretic notion induces a dual notion, defined by applying the original statement to the order dual of a given set. This exchanges ≤ and ≥, meets and joins, zero and unit.

• Extension. For partial orders ≤ and ≤′ on a setX, ≤′ is an extension of ≤ provided that for all elementsx andy ofX,x ≤y implies thatx ≤′y.

• Filter. A subsetX of a posetP is called a filter if it is a filtered upper set. The dual notion is calledideal.
• Filtered. Anon-empty subsetX of a posetP is called filtered, if, for all elementsx andy ofX, there is an elementz ofX such thatz ≤x andz ≤y. The dual notion is calleddirected.
• Finite element. Seecompact.
• Frame. A frameF is a complete lattice, in which, for everyx inF and every subsetY ofF, the infinite distributive lawx ∧${\displaystyle \bigvee }$Y =${\displaystyle \bigvee }${x ∧y |y inY} holds. Frames are also known aslocales and as completeHeyting algebras.

• Galois connection. Given two posetsP andQ, a pair of monotone functionsF:P →Q andG:Q →P is called a Galois connection, ifF(x) ≤y is equivalent tox ≤G(y), for allx inP andy inQ.F is called thelower adjoint ofG andG is called theupper adjoint ofF.
• Greatest element. For a subsetX of a posetP, an elementa ofX is called the greatest element ofX, ifx ≤a for every elementx inX. The dual notion is calledleast element.
• Ground set. The ground set of a poset (X, ≤) is the setX on which the partial order ≤ is defined.

• Heyting algebra. A Heyting algebraH is a bounded lattice in which the functionf_a:H →H, given byf_a(x) =a ∧x is the lower adjoint of aGalois connection, for every elementa ofH. The upper adjoint off_a is then denoted byg_a, withg_a(x) =a ⇒;x. EveryBoolean algebra is a Heyting algebra.
• Hasse diagram. A Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of itstransitive reduction.
• Homogeneous relation. Ahomogeneous relation on a set${\displaystyle X}$ is a subset of${\displaystyle X\times X.}$ Said differently, it is abinary relation over${\displaystyle X}$ and itself.

• Ideal. Anideal is a subsetX of a posetP that is a directed lower set. The dual notion is calledfilter.
• Incidence algebra. Theincidence algebra of a poset is theassociative algebra of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; seeincidence algebra for the details.
• Infimum. For a posetP and a subsetX ofP, the greatest element in the set of lower bounds ofX (if it exists, which it may not) is called theinfimum,meet, orgreatest lower bound ofX. It is denoted by infX or${\displaystyle \bigwedge }$X. The infimum of two elements may be written as inf{x,y} orx ∧y. If the setX is finite, one speaks of afinite infimum. The dual notion is calledsupremum.
• Interval. For two elementsa,b of a partially ordered setP, theinterval [a,b] is the subset {x inP |a ≤x ≤b} ofP. Ifa ≤b does not hold the interval will be empty.
• Interval finite poset. A partially ordered setP isinterval finite if every interval of the form {x in P | x ≤ a} is a finite set.^[2]
• Inverse. Seeconverse.
• Irreflexive. ArelationR on a setX is irreflexive, if there is no elementx inX such thatx R x.
• Isotone. Seemonotone.

• Join. Seesupremum.

• Lattice. A lattice is a poset in which all non-empty finite joins (suprema) and meets (infima) exist.
• Least element. For a subsetX of a posetP, an elementa ofX is called the least element ofX, ifa ≤x for every elementx inX. The dual notion is calledgreatest element.
• Thelength of a chain is the number of elements less one. A chain with 1 element has length 0, one with 2 elements has length 1, etc.
• Linear. Seetotal order.
• Linear extension. A linear extension of a partial order is an extension that is a linear order, or total order.
• Locale. A locale is acomplete Heyting algebra. Locales are also calledframes and appear inStone duality andpointless topology.
• Locally finite poset. A partially ordered setP islocally finite if every interval [a,b] = {x inP |a ≤x ≤b} is a finite set.
• Lower bound. A lower bound of a subsetX of a posetP is an elementb ofP, such thatb ≤x, for allx inX. The dual notion is calledupper bound.
• Lower set. A subsetX of a posetP is called a lower set if, for all elementsx inX andp inP,p ≤x implies thatp is contained inX. The dual notion is calledupper set.

• Maximal chain. Achain in a poset to which no element can be added without losing the property of being totally ordered. This is stronger than being a saturated chain, as it also excludes the existence of elements either less than all elements of the chain or greater than all its elements. A finite saturated chain is maximal if and only if it contains both a minimal and a maximal element of the poset.
• Maximal element. A maximal element of a subsetX of a posetP is an elementm ofX, such thatm ≤x impliesm =x, for allx inX. The dual notion is calledminimal element.
• Maximum element. Synonym of greatest element. For a subsetX of a posetP, an elementa ofX is called the maximum element ofX ifx ≤a for every elementx inX. A maximum element is necessarily maximal, but the converse need not hold.
• Meet. Seeinfimum.
• Minimal element. A minimal element of a subsetX of a posetP is an elementm ofX, such thatx ≤m impliesm =x, for allx inX. The dual notion is calledmaximal element.
• Minimum element. Synonym of least element. For a subsetX of a posetP, an elementa ofX is called the minimum element ofX ifx ≥a for every elementx inX. A minimum element is necessarily minimal, but the converse need not hold.
• Monotone. A functionf between posetsP andQ is monotone if, for all elementsx,y ofP,x ≤y (inP) impliesf(x) ≤f(y) (inQ). Other names for this property areisotone andorder-preserving. Inanalysis, in the presence oftotal orders, such functions are often calledmonotonically increasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is calledantitone ororder reversing.

• Order-dual. The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse.
• Order-embedding. A functionf between posetsP andQ is an order-embedding if, for all elementsx,y ofP,x ≤y (inP) is equivalent tof(x) ≤f(y) (inQ).
• Order isomorphism. A mappingf:P →Q between two posetsP andQ is called an order isomorphism, if it isbijective and bothf andf⁻¹ aremonotone functions. Equivalently, an order isomorphism is a surjectiveorder embedding.
• Order-preserving. Seemonotone.
• Order-reversing. Seeantitone.

• Partial order. A partial order is abinary relation that isreflexive,antisymmetric, andtransitive. In a slight abuse of terminology, the term is sometimes also used to refer not to such a relation, but to its corresponding partially ordered set.
• Partially ordered set. A partially ordered set${\displaystyle (P,\le ),}$ orposet for short, is a set${\displaystyle P}$ together with a partial order${\displaystyle \le }$ on${\displaystyle P.}$
• Poset. A partially ordered set.
• Preorder. A preorder is abinary relation that isreflexive andtransitive. Such orders may also be calledquasiorders ornon-strict preorder. The termpreorder is also used to denote anacyclicbinary relation (also called anacyclic digraph).
• Preordered set. A preordered set${\displaystyle (P,\le )}$ is a set${\displaystyle P}$ together with a preorder${\displaystyle \le }$ on${\displaystyle P.}$
• Preserving. A functionf between posetsP andQ is said to preserve suprema (joins), if, for all subsetsX ofP that have a supremum supX inP, we find that sup{f(x):x inX} exists and is equal tof(supX). Such a function is also calledjoin-preserving. Analogously, one says thatf preserves finite, non-empty, directed, or arbitrary joins (or meets). The converse property is calledjoin-reflecting.
• Prime. AnidealI in a latticeL is said to be prime, if, for all elementsx andy inL,x ∧y inI impliesx inI ory inI. The dual notion is called aprime filter. Equivalently, a set is a prime filterif and only if its complement is a prime ideal.
• Principal. A filter is calledprincipal filter if it has a least element. Dually, aprincipal ideal is an ideal with a greatest element. The least or greatest elements may also be calledprincipal elements in these situations.
• Projection (operator). A self-map on apartially ordered set that ismonotone andidempotent underfunction composition. Projections play an important role indomain theory.
• Pseudo-complement. In aHeyting algebra, the elementx ⇒;0 is called the pseudo-complement ofx. It is also given by sup{y :y ∧x = 0}, i.e. as the least upper bound of all elementsy withy ∧x = 0.

• Quasiorder. Seepreorder.
• Quasitransitive. A relation is quasitransitive if the relation on distinct elements is transitive. Transitive implies quasitransitive and quasitransitive implies acyclic.^[1]

• Reflecting. A functionf between posetsP andQ is said to reflect suprema (joins), if, for all subsetsX ofP for which the supremum sup{f(x):x inX} exists and is of the formf(s) for somes inP, then we find that supX exists and that supX =s . Analogously, one says thatf reflects finite, non-empty, directed, or arbitrary joins (or meets). The converse property is calledjoin-preserving.
• Reflexive. Abinary relationR on a setX is reflexive, ifx R x holds for every elementx inX.
• Residual. A dual map attached to aresiduated mapping.
• Residuated mapping. A monotone map for which the preimage of a principal down-set is again principal. Equivalently, one component of a Galois connection.

• Saturated chain. Achain in a poset such that no element can be addedbetween two of its elements without losing the property of being totally ordered. If the chain is finite, this means that in every pair of successive elements the larger one covers the smaller one. See also maximal chain.
• Scattered. A total order is scattered if it has no densely ordered subset.
• Scott-continuous. A monotone functionf :P →Q between posetsP andQ is Scott-continuous if, for every directed setD that has a supremum supD inP, the set {fx |x inD} has the supremumf(supD) inQ. Stated differently, a Scott-continuous function is one that preserves all directed suprema. This is in fact equivalent to beingcontinuous with respect to theScott topology on the respective posets.
• Scott domain. A Scott domain is a partially ordered set which is abounded completealgebraiccpo.
• Scott open. SeeScott topology.
• Scott topology. For a posetP, a subsetO isScott-open if it is anupper set and all directed setsD that have a supremum inO have non-empty intersection withO. The set of all Scott-open sets forms atopology, theScott topology.
• Semilattice. A semilattice is a poset in which either all finite non-empty joins (suprema) or all finite non-empty meets (infima) exist. Accordingly, one speaks of ajoin-semilattice ormeet-semilattice.
• Smallest element. Seeleast element.
• Sperner property of a partially ordered set
• Sperner poset
• Strictly Sperner poset
• Strongly Sperner poset
• Strict order. Seestrict partial order.
• Strict partial order. A strict partial order is ahomogeneous binary relation that istransitive,irreflexive, andantisymmetric.
• Strict preorder. Seestrict partial order.
• Supremum. For a posetP and a subsetX ofP, theleast element in the set ofupper bounds ofX (if it exists, which it may not) is called thesupremum,join, orleast upper bound ofX. It is denoted by supX or${\displaystyle \bigvee }$X. The supremum of two elements may be written as sup{x,y} orx ∨y. If the setX is finite, one speaks of afinite supremum. The dual notion is calledinfimum.
• Suzumura consistency. A binary relation R is Suzumura consistent ifx R^∗y implies thatx Ry or noty Rx.^[1]
• Symmetric relation. Ahomogeneous relationR on a setX is symmetric, ifx R y impliesy R x, for all elementsx,y inX.

• Top. Seeunit.
• Total order. A total orderT is a partial order in which, for eachx andy inT, we havex ≤y ory ≤x. Total orders are also calledlinear orders orchains.
• Total relation. Synonym forConnected relation.
• Transitive relation. ArelationR on a setX is transitive, ifx R y andy R z implyx R z, for all elementsx,y,z inX.
• Transitive closure. The transitive closure R^∗ of a relation R consists of all pairsx,y for which there exists a finite chainx Ra,a Rb, ...,z Ry.^[1]

• Unit. Thegreatest element of a posetP can be calledunit or just1 (if it exists). Another common term for this element istop. It is the infimum of the empty set and the supremum ofP. The dual notion is calledzero.
• Up-set. Seeupper set.
• Upper bound. An upper bound of a subsetX of a posetP is an elementb ofP, such thatx ≤b, for allx inX. The dual notion is calledlower bound.
• Upper set. A subsetX of a posetP is called an upper set if, for all elementsx inX andp inP,x ≤p implies thatp is contained inX. The dual notion is calledlower set.

• Valuation. Given a lattice${\displaystyle X}$, a valuation${\displaystyle \nu :X\to [0,1]}$ is strict (that is,${\displaystyle \nu (\varnothing )=0}$), monotone, modular (that is,${\displaystyle \nu (U)+\nu (V)=\nu (U\cup V)+\nu (U\cap V)}$) and positive. Continuous valuations are a generalization of measures.

• Way-below relation. In a posetP, some elementx isway belowy, writtenx<<y, if for all directed subsetsD ofP which have a supremum,y ≤sup D impliesx ≤d for somed inD. One also says thatxapproximatesy. See alsodomain theory.
• Weak order. A partial order ≤ on a setX is a weak order provided that the poset (X, ≤) isisomorphic to a countable collection of sets ordered by comparison ofcardinality.

• Zero. Theleast element of a posetP can be calledzero or just0 (if it exists). Another common term for this element isbottom. Zero is the supremum of the empty set and the infimum ofP. The dual notion is calledunit.

The definitions given here are consistent with those that can be found in the following standard reference books:

• B. A. Davey and H. A. Priestley,Introduction to Lattices and Order, 2nd Edition, Cambridge University Press, 2002.
• G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott,Continuous Lattices and Domains, InEncyclopedia of Mathematics and its Applications, Vol. 93, Cambridge University Press, 2003.

Specific definitions:

• Deng, Bangming (2008),Finite dimensional algebras and quantum groups, Mathematical surveys and monographs, vol. 150, American Mathematical Society,ISBN 978-0-8218-4186-0

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