This is a glossary of some terms used in various branches of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 that are related to the fields of order

Order theory

Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

, lattice

Lattice (order)

In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

, and domain theory

Domain theory

Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...

. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:

• completeness properties
  Completeness (order theory)
  In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
  
   of partial orders
• distributivity laws
  Distributivity (order theory)
  In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima...
  
   of order theory
• preservation properties of functions between posets.

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, ≤ will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the strict order induced by ≤.

A

• Acyclic. A binary relation
  Binary relation
  In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
  
   is acyclic if it contains no "cycles": equivalently, its transitive closure
  Transitive closure
  In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such that R+ contains R and R+ is minimal . If the binary relation itself is transitive, then the transitive closure will be that same binary relation; otherwise, the transitive closure...
  
   is antisymmetric
  Antisymmetric
  The word antisymmetric refers to a change to an opposite quantity when another quantity is symmetrically changed. This concept is related to that of Symmetry and Asymmetry. The difference between these three concepts can be simply illustrated with Latin letters. The character "A" is symmetric about...
  
  .

• Adjoint. See Galois connection.

• Alexandrov topology
  Alexandrov topology
  In topology, an Alexandrov space is a topological space in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open...
  
  . For a preordered set P, any upper set O is Alexandrov-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
  Antichain
  In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two elements in the subset are incomparable. Let S be a partially ordered set...
  
  . An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation.

• Approximates relation. See way-below relation.

• A relation
  Relation (mathematics)
  In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
  
   R on a set X is antisymmetric
  Antisymmetric relation
  In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
  
  , if x R y and y R x implies x = y, for all elements x, y in X.

• An antitone function f between posets P and Q is a function for which, for all elements x, y of P, x ≤ y (in P) implies f(y) ≤ f(x) (in Q). Another name for this property is order-reversing. In analysis
  Mathematical analysis
  Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
  
  , in the presence of total order
  Total order
  In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
  
  s, such functions are often called monotonically decreasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is called monotone or order-preserving.

• Asymmetric
  Asymmetric relation
  Asymmetric often means, simply: not symmetric. In this sense an asymmetric relation is a binary relation which is not a symmetric relation.That is,\lnot....
  
  . A relation
  Relation (mathematics)
  In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
  
   R on a set X is asymmetric, if x R y implies not y R x, for all elements x, y in X.

• An atom in a poset P with least element 0, is an element that is minimal among all elements that are unequal to 0.

• A atomic poset P with least element 0 is one in which, for every non-zero element x of P, there is an atom a of P with a ≤ x.

B

• Base. See continuous poset.

• A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1.

• A bounded poset is one that has a least element and a greatest element.

• A poset is bounded complete if every of its subsets with some upper bound also has a least such upper bound. The dual notion is not common.

C

• Chain. A chain is a totally ordered set or a totally ordered subset of a poset. See also total order.

• Closure operator
  Closure operator
  In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
  
  . A closure operator on the poset P is a function C : P → P that is monotone, idempotent, and satisfies C(x) ≥ x for all x in P.

• Compact
  Compact element
  In the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element....
  
  . An element x of a poset is compact if it is way below itself, i.e. x<<x. One also says that such an x is finite.

• Comparable. Two elements x and y of a poset P are comparable if either x ≤ y or y ≤ x.

• Comparability graph
  Comparability graph
  In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order...
  
  . The comparability graph of a poset (P, ≤) is the graph
  Graph (mathematics)
  In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
  
   with vertex set P in which the edges are those pairs of distinct elements of P that are comparable under ≤ (and, in particular, under its reflexive reduction <).

• A complete Boolean algebra
  Complete Boolean algebra
  In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum . Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing...
  
  is a Boolean algebra that is a complete lattice.

• Complete Heyting algebra
  Complete Heyting algebra
  In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...
  
  . A Heyting algebra
  Heyting algebra
  In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
  
   that is a complete lattice is called a complete Heyting algebra. This notion coincides with the concepts frame and locale.

• Complete lattice
  Complete lattice
  In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
  
  . A complete lattice
  Lattice (order)
  In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
  
   is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.

• Complete partial order
  Complete partial order
  In mathematics, directed complete partial orders and ω-complete partial orders are special classes of partially ordered sets, characterized by particular completeness properties...
  
  . A complete partial order, or cpo, is a directed complete partial order (q.v.) with least element.

• Complete semilattice. The notion of a complete semilattice is defined in different ways. As explained in the article on completeness (order theory)
  Completeness (order theory)
  In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
  
  , 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 be bounded complete cpos
  Complete partial order
  In mathematics, directed complete partial orders and ω-complete partial orders are special classes of partially ordered sets, characterized by particular completeness properties...
  
  , which is arguably the most complete class of posets that are not already complete lattices.

• Completely distributive lattice
  Completely distributive lattice
  In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets....
  
  . A complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets.

• Completion. A completion of a poset is an order-embedding
  Order-embedding
  In mathematical order theory, an order-embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order-embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism...
  
   of the poset in a complete lattice.

• Continuous poset. A poset is continuous if it has a base, i.e. a subset B of P such that every element x of P is the supremum of a directed set contained in {y in B | y<<x}.

• Continuous function. See Scott-continuous.

• Converse. The converse <° of an order < is that in which x <° y whenever y < x.

• Cover. An element y of a poset P is said to cover an element x of P (and is called a cover of x) if x < y and there is no element z of P such that x < z < y.

• cpo
  CPO
  -Officers:* Chief Performance Officer of the United States* Chief Petty Officer, a military rank* Chief privacy officer, an executive responsible for managing issues of privacy laws and policies...
  
  . See complete partial order.

D

• dcpo. See directed complete partial order.

• A dense
  Dense order
  In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x...
  
  poset P is one in which, for all elements x and y in P with x < y, there is an element z in P, such that x < z < y. A subset Q of P is dense in P if for any elements x < y in P, there is an element z in Q such that x < z < y.

• Directed
  Directed set
  In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
  
  . A non-empty subset X of a poset P is called directed, if, for all elements x and y of X, there is an element z of X such that x ≤ z and y ≤ z. The dual notion is called filtered.

• Directed complete partial order. A poset D is said to be a directed complete poset, or dcpo, if every directed subset of D has a supremum.

• Distributive
  Distributivity (order theory)
  In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima...
  
  . A lattice L is called distributive if, for all x, y, and z in L, we find that x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). This condition is known to be equivalent to its order dual. A meet-semilattice
  Semilattice
  In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
  
   is distributive if for all elements a, b and x, a ∧ b ≤ x implies the existence of elements a' ≥ a and b' ≥ b such that a' ∧ b' = x. See also completely distributive.

• Domain
  Domain theory
  Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...
  
  . Domain is a general term for objects like those that are studied in domain theory
  Domain theory
  Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...
  
  . If used, it requires further definition.

• Down-set. See lower set.

• Dual
  Duality (order theory)
  In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
  
  . For a poset (P, ≤), the dual order P^d = (P, ≥) is defined by setting x ≥ y if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
   y ≤ x. The dual order of P is sometimes denoted by P^op, and is also called opposite or converse 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.

E

• Extension. For partial orders ≤ and ≤’ on a set X, ≤’ is an extension of ≤ provided that for all elements x and y of X, x ≤ y implies that x ≤’ y.

F

• Filter
  Filter (mathematics)
  In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
  
  . A subset X of a poset P is called a filter if it is a filtered upper set. The dual notion is called ideal.

• Filtered. A non-empty subset X of a poset P is called filtered, if, for all elements x and y of X, there is an element z of X such that z ≤ x and z ≤ y. The dual notion is called directed.

• Finite element. See compact.

• Frame
  Complete Heyting algebra
  In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...
  
  . A frame F is a complete lattice, in which, for every x in F and every subset Y of F, the infinite distributive law x ∧ Y = {x ∧ y | y in Y} holds. Frames are also known as locales and as complete Heyting algebra
  Heyting algebra
  In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
  
  s.

G

• Galois connection
  Galois connection
  In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
  
  . Given two posets P and Q, a pair of monotone functions F:P → Q and G:Q → P is called a Galois connection, if F(x) ≤ y is equivalent to x ≤ G(y), for all x in P and y in Q. F is called the lower adjoint of G and G is called the upper adjoint of F.

• Greatest element
  Greatest element
  In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
  
  . For a subset X of a poset P, an element a of X is called the greatest element of X, if x ≤ a for every element x in X. The dual notion is called least element.

• Ground set. The ground set of a poset (X, ≤) is the set X on which the partial order ≤ is defined.

H

• Heyting algebra
  Heyting algebra
  In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
  
  . A Heyting algebra H is a bounded lattice in which the function f_a: H → H, given by f_a(x) = a ∧ x is the lower adjoint of a Galois connection
  Galois connection
  In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
  
  , for every element a of H. The upper adjoint of f_a is then denoted by g_a, with g_a(x) = a ⇒ x. Every Boolean algebra is a Heyting algebra.
• Hasse diagram
  Hasse diagram
  In order theory, a branch of mathematics, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction...
  
  . A Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction
  Transitive reduction
  In mathematics, a transitive reduction of a binary relation R on a set X is a minimal relation R' on X such that the transitive closure of R' is the same as the transitive closure of R. If the transitive closure of R is antisymmetric and finite, then R' is unique...
  
  .

I

• An ideal
  Ideal (order theory)
  In mathematical order theory, an ideal is a special subset of a partially ordered set . Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion...
  
  is a subset X of a poset P that is a directed lower set. The dual notion is called filter.

• The incidence algebra
  Incidence algebra
  In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for any locally finite partially ordered setand commutative ring with unity.-Definition:...
  
  of a poset is the associative algebra
  Associative algebra
  In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
  
   of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; see incidence algebra
  Incidence algebra
  In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for any locally finite partially ordered setand commutative ring with unity.-Definition:...
  
   for the details.

• Infimum
  Infimum
  In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
  
  . For a poset P and a subset X of P, the greatest element in the set of lower bounds of X (if it exists, which it may not) is called the infimum, meet, or greatest lower bound of X. It is denoted by inf X or X. The infimum of two elements may be written as inf{x,y} or x ∧ y. If the set X is finite, one speaks of a finite infimum. The dual notion is called supremum.

• Interval
  Interval (mathematics)
  In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
  
  . For two elements a, b of a partially ordered set P, the interval [a,b] is the subset {x in P | a ≤ x ≤ b} of P. If a ≤ b does not hold the interval will be empty.

• Interval finite poset. A partially ordered set P is interval finite if every interval of the form {x in P | x ≤ a} is a finite set.

• Inverse. See converse.

• Irreflexive. A relation
  Relation (mathematics)
  In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
  
   R on a set X is irreflexive, if there is no element x in X such that x R x.

• Isotone. See monotone.

L

• Lattice
  Lattice (order)
  In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
  
  . A lattice is a poset in which all non-empty finite joins (suprema) and meets (infima) exist.

• Least element. For a subset X of a poset P, an element a of X is called the least element of X, if a ≤ x for every element x in X. The dual notion is called greatest element.

• The length 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. See total order.

• Linear extension
  Linear extension
  In order theory, a branch of mathematics, a linear extension of a partial order is a linear order that is compatible with the partial order.-Definitions:...
  
  . A linear extension of a partial order is an extension that is a linear order, or total order.

• Locale
  Complete Heyting algebra
  In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...
  
  . A locale is a complete Heyting algebra. Locales are also called frames and appear in Stone duality
  Stone duality
  In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation...
  
   and pointless topology
  Pointless topology
  In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...
  
  .

• Locally finite poset
  Locally finite poset
  In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements....
  
  . A partially ordered set P is locally finite if every interval [a, b] = {x in P | a ≤ x ≤ b} is a finite set.

• Lower bound. A lower bound of a subset X of a poset P is an element b of P, such that b ≤ x, for all x in X. The dual notion is called upper bound.

• Lower set. A subset X of a poset P is called a lower set if, for all elements x in X and p in P, p ≤ x implies that p is contained in X. The dual notion is called upper set.

M

• Maximal chain. A chain 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
  Maximal element
  In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually...
  
  . A maximal element of a subset X of a poset P is an element m of X, such that m ≤ x implies m = x, for all x in X. The dual notion is called minimal element.

• Meet. See infimum.

• Minimal element. A minimal element of a subset X of a poset P is an element m of X, such that x ≤ m implies m = x, for all x in X. The dual notion is called maximal element.

• Monotone. A function f between posets P and Q is monotone if, for all elements x, y of P, x ≤ y (in P) implies f(x) ≤ f(y) (in Q). Other names for this property are isotone and order-preserving. In analysis
  Mathematical analysis
  Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
  
  , in the presence of total order
  Total order
  In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
  
  s, such functions are often called monotonically increasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is called antitone or order reversing.

O

• 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
  Order-embedding
  In mathematical order theory, an order-embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order-embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism...
  
  . A function f between posets P and Q is an order-embedding if, for all elements x, y of P, x ≤ y (in P) is equivalent to f(x) ≤ f(y) (in Q).

• Order isomorphism
  Order isomorphism
  In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets . Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of...
  
  . A mapping f: P → Q between two posets P and Q is called an order isomorphism, if it is bijective and both f and f^-1 are monotone. Equivalently, an order isomorphism is a surjective order embedding.

• Order-preserving. See monotone.

• Order-reversing. See antitone.

P

• Partial order. A partial order is a binary relation
  Binary relation
  In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
  
   that is reflexive
  Reflexive relation
  In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
  
  , antisymmetric
  Antisymmetric relation
  In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
  
  , and transitive
  Transitive relation
  In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
  
  . 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
  Partially ordered set
  In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
  
  . A partially ordered set (P, ≤), or poset for short, is a set P together with a partial order ≤ on P.

• Poset. A partially ordered set.

• Preorder
  Preorder
  In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
  
  . A preorder is a binary relation
  Binary relation
  In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
  
   that is reflexive
  Reflexive relation
  In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
  
   and transitive
  Transitive relation
  In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
  
  . Such orders may also be called quasiorders. The term preorder is also used to denote an acyclic
  Acyclic
  Acyclic can refer to:* In chemistry, a compound which is not cyclic, e.g. alkanes and acyclic aliphatic compounds* In mathematics:** A graph without a cycle, especially*** A directed acyclic graph...
  
   binary relation
  Binary relation
  In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
  
   (also called an acyclic digraph).

• Preserving
  Limit-preserving function (order theory)
  In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima. Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of the set...
  
  . A function f between posets P and Q is said to preserve suprema (joins), if, for all subsets X of P that have a supremum sup X in P, we find that sup{f(x): x in X} exists and is equal to f(sup X). Such a function is also called join-preserving. Analogously, one says that f preserves finite, non-empty, directed, or arbitrary joins (or meets). The converse property is called join-reflecting.

• Prime. An ideal I in a lattice L is said to be prime, if, for all elements x and y in L, x ∧ y in I implies x in I or y in I. The dual notion is called a prime filter. Equivalently, a set is a prime filter if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
   its complement is a prime ideal.

• Principal. A filter is called principal filter if it has a least element. Dually, a principal ideal is an ideal with a greatest element. The least or greatest elements may also be called principal elements in these situations.

• Projection (operator). A self-map on a partially ordered set that is monotone
  Monotone
  Monotone refers to a sound, for example speech or music, that has a single unvaried tone.Monotone or monotonicity may also refer to:*Monotone , an open source revision control system*Monotone class theorem, in measure theory...
  
   and idempotent under function composition
  Function composition
  In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
  
  . Projections play an important role in domain theory
  Domain theory
  Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...
  
  .

• Pseudo-complement. In a Heyting algebra
  Heyting algebra
  In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
  
  , the element x ⇒ 0 is called the pseudo-complement of x. It is also given by sup{y : y ∧ x = 0}, i.e. as the least upper bound of all elements y with y ∧ x = 0.

R

• Reflecting. A function f between posets P and Q is said to reflect suprema (joins), if, for all subsets X of P for which the supremum sup{f(x): x in X} exists and is of the form f(s) for some s in P, then we find that sup X exists and that sup X = s . Analogously, one says that f reflects finite, non-empty, directed, or arbitrary joins (or meets). The converse property is called join-preserving.

• Reflexive
  Reflexive relation
  In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
  
  . A binary relation
  Binary relation
  In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
  
   R on a set X is reflexive, if x R x holds for all elements x, y in X.

• Residual. A dual map attached to a residuated mapping
  Residuated mapping
  In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function....
  
  .

• Residuated mapping
  Residuated mapping
  In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function....
  
  . A monotone map for which the preimage of a principal down-set is again principal. Equivalently, one component of a Galois connection.

S

• Saturated chain. A chain such that no element can be added between 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
  Scattered order
  In mathematics, specifically order theory, a scattered order is a linear order that contains no densely ordered subset with more than 1 element....
  
  . A total order is scattered if it has no densely ordered subset.

• Scott-continuous. A monotone function f : P → Q between posets P and Q is Scott-continuous if, for every directed set D that has a supremum sup D in P, the set {fx | x in D} has the supremum f(sup D) in Q. Stated differently, a Scott-continuous function is one that preserves all directed suprema. This is in fact equivalent to being continuous with respect to the Scott topology on the respective posets.

• Scott domain
  Scott domain
  In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded complete cpo. It has been named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory...
  
  . A Scott domain is a partially ordered set which is a bounded complete algebraic cpo
  Complete partial order
  In mathematics, directed complete partial orders and ω-complete partial orders are special classes of partially ordered sets, characterized by particular completeness properties...
  
  .

• Scott open. See Scott topology.

• Scott topology. For a poset P, a subset O is Scott-open if it is an upper set
  Upper set
  In mathematics, an upper set of a partially ordered set is a subset U with the property that x is in U and x≤y imply y is in U....
  
   and all directed sets D that have a supremum in O have non-empty intersection with O. The set of all Scott-open sets forms a topology
  Topology
  Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
  
  , the Scott topology.

• Semilattice
  Semilattice
  In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
  
  . 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 a join-semilattice or meet-semilattice.

• Smallest element. See least element.

• Sperner property of a partially ordered set
  Sperner property of a partially ordered set
  In the order-theoretic mathematics, a graded partially ordered set is said to have the Sperner property , if no antichain within it is larger than the largest rank level in the poset.Since every rank level is itself an antichain, the Sperner property is equivalently the property that some rank...
  
  
• Sperner poset
• Strictly Sperner poset
• Strongly Sperner poset

• Strict order. A strict order is a binary relation
  Binary relation
  In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
  
   that is antisymmetric
  Antisymmetric relation
  In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
  
  , transitive
  Transitive relation
  In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
  
  , and irreflexive.

• Supremum
  Supremum
  In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
  
  . For a poset P and a subset X of P, the least element in the set of upper bound
  Upper bound
  In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
  
  s of X (if it exists, which it may not) is called the supremum, join, or least upper bound of X. It is denoted by sup X or X. The supremum of two elements may be written as sup{x,y} or x ∨ y. If the set X is finite, one speaks of a finite supremum. The dual notion is called infimum.

• Symmetric
  Symmetric relation
  In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
  
  . A relation
  Relation (mathematics)
  In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
  
   R on a set X is symmetric, if x R y implies y R x, for all elements x, y in X.

T

• Top. See unit.

• Total order
  Total order
  In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
  
  . A total order T is a partial order in which, for each x and y in T, we have x ≤ y or y ≤ x. Total orders are also called linear orders or chains.

• Transitive
  Transitive relation
  In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
  
  . A relation
  Relation (mathematics)
  In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
  
   R on a set X is transitive, if x R y and y R z imply x R z, for all elements x, y, z in X.

U

• Unit. The greatest element
  Greatest element
  In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
  
   of a poset P can be called unit or just 1 (if it exists). Another common term for this element is top. It is the infimum of the empty set and the supremum of P. The dual notion is called zero.

• Up-set. See upper set.

• Upper bound
  Upper bound
  In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
  
  . An upper bound of a subset X of a poset P is an element b of P, such that x ≤ b, for all x in X. The dual notion is called lower bound.

• Upper set
  Upper set
  In mathematics, an upper set of a partially ordered set is a subset U with the property that x is in U and x≤y imply y is in U....
  
  . A subset X of a poset P is called an upper set if, for all elements x in X and p in P, x ≤ p implies that p is contained in X. The dual notion is called lower set.

V

• Valuation. Given a lattice , a valuation is strict (i.e., ), monotone, modular (i.e., ) and positive. Continuous valuations are a generalization of measures.

W

• Way-below relation. In a poset P, some element x is way below y, written x<<y, if for all directed subsets D of P which have a supremum, y ≤ sup D implies x ≤ d for some d in D. One also says that x approximates y. See also domain theory
  Domain theory
  Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...
  
  .

• Weak order. A partial order ≤ on a set X is a weak order provided that the poset (X, ≤) is isomorphic to a countable collection of sets ordered by comparison of cardinality.

Z

• Zero. The least element of a poset P can be called zero or just 0 (if it exists). Another common term for this element is bottom. Zero is the supremum of the empty set and the infimum of P. The dual notion is called unit.