Order theory glossary
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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:
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 Alexandrovopen. 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 waybelow relation.
 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 orderreversing. 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 nontotal orders. The dual notion is called monotone or orderpreserving.
 Asymmetric. A relation 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 nonzero 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. 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.
 Comparability graph. The comparability graph of a poset (P, ≤) is the graph 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 <).
 Complete lattice. A complete lattice is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.
 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.
 Completion. A completion of a poset is an orderembedding 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 Scottcontinuous.
 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. 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 nonempty 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.
 Distributive. 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 meetsemilattice 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.
 Dual. For a poset (P, ≤), the dual order P^{d} = (P, ≥) is defined by setting x ≥ y if and only if 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. 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 nonempty 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 ∧ Y
= {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.
 Ground set. The ground set of a poset (X, ≤) is the set X on which the partial order ≤ is defined.
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.
 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
L
 Lattice. A lattice is a poset in which all nonempty 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 extension. A linear extension of a partial order is an extension that is a linear order, or total order.
 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 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. 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.
 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 orderpreserving. 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 nontotal orders. The dual notion is called antitone or order reversing.
O
 Orderdual. The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse.
 Orderembedding. A function f between posets P and Q is an orderembedding 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.
P
 Partially ordered set. 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.
 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 joinpreserving. Analogously, one says that f preserves finite, nonempty, directed, or arbitrary joins (or meets). The converse property is called joinreflecting.
 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 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.
 Pseudocomplement. In a Heyting algebra, the element x ⇒ 0 is called the pseudocomplement 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, nonempty, directed, or arbitrary joins (or meets). The converse property is called joinpreserving.
 Residuated mapping. A monotone map for which the preimage of a principal downset 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.
 Scottcontinuous. A monotone function f : P → Q between posets P and Q is Scottcontinuous 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 Scottcontinuous 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 open. See Scott topology.
 Scott topology. For a poset P, a subset O is Scottopen if it is an upper set and all directed sets D that have a supremum in O have nonempty intersection with O. The set of all Scottopen sets forms a topology, the Scott topology.
 Semilattice. A semilattice is a poset in which either all finite nonempty joins (suprema) or all finite nonempty meets (infima) exist. Accordingly, one speaks of a joinsemilattice or meetsemilattice.
 Smallest element. See least element.
 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 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. 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
 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.
 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.
V
 Valuation. Given a lattice X,
a valuation
is strict (i.e., ),
monotone, modular (i.e., )
and positive. Continuous valuations are a generalization
of measures.
W
 Waybelow 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.
 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.
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.
Published  March 2009
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