Glossary of order theory Guide, Meaning , Facts, Information and Description

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:

• 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 ≤.

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.

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.

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.

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.

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.

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.

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.

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 convoluation; 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.

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.

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.

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 &rarr 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 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.

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.

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.

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.

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, p ≤ x implies that p is contained in X. The dual notion is called lower set.

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.

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.

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.

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