This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles: Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ... This is a list of order topics, by Wikipedia page. ...

• completeness properties of partial orders
• distributivity laws 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 ≤. In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ... 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. ... In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i. ... A strict weak ordering is a binary relation that defines an equivalence relation and has the properties stated below. ...

A

• Adjoint. See Galois connection.

• Alexandrov topology. 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. 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 R on a set X is antisymmetric, 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, in the presence of total orders, 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.

• An asymmetric relation R is a relation that is not symmetric.

• 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.

In general topology the open sets of a topological space satisfy by definition the conditions: The union of arbitrarily many open sets is open. ... Let S be a partially ordered set. ... In mathematics, an n-ary relation (or n-place relation or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... Look up Analysis in Wiktionary, the free dictionary An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation and relational algebra. ...

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 v ¬x = 1.

• A bounded poset is one that has a least element 0 and a greatest element 1.

• 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.

Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets which have some upper bound also have a least upper bound. ...

C

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

• Closure operator. 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. 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.

• A complete Boolean algebra is a Boolean algebra that is a complete lattice.

• Complete Heyting algebra. A Heyting algebra that is a complete lattice is called a complete Heyting algebra. This notion coincides with the concepts frame and locale.

• Complete lattice. A complete lattice is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.

• Complete partial order. A complete partial order, or cpo, is a directed complete poset 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), 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, which is arguably the most complete class of posets that are not already complete lattices.

• Completely distributive. A complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets. For the formal statement see the article on distributivity (order theory). The concept of complete distributivity is self-dual.

• 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.

• cpo. See complete partial order.

In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y, then C(x) ≤ C(y), i. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... 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 directed set that does not already contain members above the compact element. ... In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ... The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ... In mathematics, directed complete partial orders and complete partial orders are special classes of partially ordered sets. ... In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ... In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets which have some upper bound also have a least upper bound. ... In mathematics, directed complete partial orders and complete partial orders are special classes of partially ordered sets. ... 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. ... CPO can refer to: Calgary Philharmonic Orchestra Certified Pre-Owned - a qualification for a used vehicle sold by dealers. ...

D

• dcpo. See directed complete partial order.

• A dense 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. 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. A lattice L is called distributive if, for all x, y, and z in L, we find that x ^ (y v z) = (x ^ y) v (x ^ z). This condition is known to be equivalent to its order dual. A meet-semilattice 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 is a general term for objects like those that are studied in domain theory. If used, it requires further definition.

• Down-set. See lower set.

• Dual. For a poset (P, ≤), the dual order (P, ≥) is defined by setting x ≥ y iff 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.

In mathematics, directed complete partial orders and complete partial orders are special classes of partially ordered sets. ... In mathematics, the term dense has at least three different meanings. ... In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there exists a c in A... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In mathematics, directed complete partial orders and complete partial orders are special classes of partially ordered sets. ... 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. ... In mathematical order theory, a semilattice is a partially ordered set (poset) within which either all binary sets have either a supremum (join) or all binary sets have an infimum (meet). ... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ... In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

F

• Filter. 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. 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 ^ VY = V{x ^ y | y in Y} holds. Frames are also known as locales and as complete Heyting algebras.

In mathematics, a filter is a special subset of a partially ordered set. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ...

G

• Galois connection. 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. 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.

In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ... 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. ...

H

• Heyting algebra. 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, 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.

In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...

I

• An ideal is a subset X of a poset P that is a directed lower set. The dual notion is called filter.

• The incidence algebra of a poset is the associative algebra of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; see incidence algebra for the details.

• Infimum. 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. 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.

• Irreflexive. A relation R on a set X is irreflexive, if there is no element x in X such that x R x.

In mathematical order theory, an ideal is a special subset of a partially ordered set. ... In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In logic and mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself. ... In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation and relational algebra. ...

J

• Join. See supremum.

L

• Lattice. 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.

• Linear. See total order.

• Locale. A locale is a complete Heyting algebra. Locales are also called frames and appear in Stone duality and pointless topology.

• Locally finite poset. 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.

The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ... 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. ... In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. ... In mathematics, especially in topology and order theory, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. ... Pointless topology is an approach to topology which avoids the mentioning of points. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ...

M

• Maximal element. 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). Another name for this property is order-preserving. In analysis, in the presence of total orders, 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.

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. ... 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. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... Look up Analysis in Wiktionary, the free dictionary An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

O

• Order-embedding. 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. 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.

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. ... 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. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...

P

• Partial order. A partial order is a binary relation that is reflexive, antisymmetric, and transitive. Since both notions depend on each other, the term is also used to refer to a partially ordered set.

• Partially ordered set. A partially ordered set, or poset for short, is a set P together with a partial order ≤ defined on P.

• poset. See partial order.

• Preorder. A preorder is a binary relation that is reflexive and transitive. Such orders may also be called quasiorders.

• Preserving. 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 iff 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.

• Pseudo-complement. In a Heyting algebra, 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.

In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In logic and mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... This article is about the mathematics concept. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In logic and mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i. ... In mathematical order theory, an ideal is a special subset of a partially ordered set. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In mathematical order theory, an ideal is a special subset of a partially ordered set. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ...

Q

• Quasiorder. See preorder.

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. A binary relation R on a set X is reflexive, if x R x holds for all elements x, y in X.

In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i. ... In logic and mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ...

S

• 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. A Scott domain is a partially ordered set which is a bounded complete algebraic cpo.

• Scott open. See Scott topology.

• Scott topology. For a poset P, an upper set O is Scott-open if 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, the Scott-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 a join-semilattice or meet-semilattice.

• Smallest element. See least element.

• Strict order. A strict order is a binary relation that is antisymmetric, transitive, and irreflexive.

• Supremum. For a poset P and a subset X of P, the least element in the set of upper bounds 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 VX. The supremum of two elements may be written as sup{x,y} or x v y. If the set X is finite, one speaks of a finite supremum. The dual notion is called infimum.

• Symmetric. A relation R on a set X is symmetric, if x R y implies y R x, for all elements x, y in X.

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... In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. ... In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded complete cpo. ... In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets which have some upper bound also have a least upper bound. ... In mathematics, directed complete partial orders and complete partial orders are special classes of partially ordered sets. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematical order theory, a semilattice is a partially ordered set (poset) within which either all binary sets have either a supremum (join) or all binary sets have an infimum (meet). ... A strict weak ordering is a binary relation that defines an equivalence relation and has the properties stated below. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In logic and mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... 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. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... 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 mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation and relational algebra. ...

T

• Top. See unit.

• Total order. 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. A relation 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.

In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation and relational algebra. ...

U

• Unit. The greatest element 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. 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. 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.

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. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In mathematics, an upper set is a subset Y of a given set X such that, for all elements x and y, if x is less than or equal to y and x is an element of Y, then y is also in Y. More formally, An upper set is...

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 is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ...

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.

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. ...

References

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, 2^nd 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, In Encyclopedia of Mathematics and its Applications, Vol. 93, Cambridge University Press, 2003.