Last modified on 5 July 2011, at 01:58

Appendix:Glossary of order theory

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 ≤.


Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z


AEdit

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 xy. In other words, the order relation of an antichain is just the identity relation.
Approximates relation 
See way-below relation.
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.
antitone 
function f between posets P and Q is a function for which, for all elements x, y of P, xy (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.
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.
atom 
In a poset P with least element 0, an element that is minimal among all elements that are unequal to 0.
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 ax.

BEdit

Base 
See continuous poset.
Boolean algebra 
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.
bounded poset 
One that has a least element 0 and a greatest element 1.
bounded complete 
Of poset: if every of its subsets with some upper bound also has a least such upper bound. The dual notion is not common.

CEdit

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 : PP 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 xy or yx.
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 lattice 
A complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets.
Completion 
A completion of a poset is an order-embedding 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.
Cover 
An element y of a poset P is said to cover an element x of P if x < y and there is no element z of P such that x < z < y.
cpo 
See complete partial order.

DEdit

dcpo 
See directed complete partial order.
dense order 
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 xz and yz. 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 ∧ (yz) = (xy) ∨ (xz). This condition is known to be equivalent to its order dual. A meet-semilattice is distributive if for all elements a, b and x, abx 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 if and only if y ≤ x. The dual order of P is sometimes denoted by Pop, 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.

FEdit

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 zx and zy. 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\bigveeY = \bigvee{xy | y in Y} holds. Frames are also known as locales and as complete Heyting algebras.

GEdit

Galois connection 
Given two posets P and Q, a pair of monotone functions F:PQ and G:QP is called a Galois connection, if F(x) ≤ y is equivalent to xG(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 xa for every element x in X. The dual notion is called least element.

HEdit

Heyting algebra 
A Heyting algebra H is a bounded lattice in which the function fa: HH, given by fa(x) = ax is the lower adjoint of a Galois connection, for every element a of H. The upper adjoint of fa is then denoted by ga, with ga(x) = ax. Every Boolean algebra is a Heyting algebra.

IEdit

ideal 
A subset X of a poset P that is a directed lower set. The dual notion is called filter.
incidence algebra 
Of a poset: 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 \bigwedgeX. The infimum of two elements may be written as inf{x,y} or xy. 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 | axb} of P. If ab 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.

JEdit

Join 
See supremum.

LEdit

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 ax 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 | axb} 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 bx, 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, px implies that p is contained in X. The dual notion is called upper set.

MEdit

Maximal element 
A maximal element of a subset X of a poset P is an element m of X, such that mx 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 xm 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, xy (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.

OEdit

Order-embedding 
A function f between posets P and Q is an order-embedding if, for all elements x, y of P, xy (in P) is equivalent to f(x) ≤ f(y) (in Q).
Order isomorphism 
A mapping f: PQ 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.

PEdit

partial order 
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 
(poset for short) A set P together with a partial order ≤ defined on P.
poset 
See partially ordered set.
preorder 
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 ideal 
An ideal I in a lattice L such that for all elements x and y in L, xy in I implies x in I or y in I.
prime filter 
A set such that its complement is a prime ideal.
principal filter 
A filter that has a least element.
principal ideal 
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 x0 is called the pseudo-complement of x. It is also given by sup{y : yx = 0}, i.e. as the least upper bound of all elements y with yx = 0.

QEdit

quasiorder 
See preorder.

REdit

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.

SEdit

Scott-continuous 
A monotone function f : PQ 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 \bigveeX. The supremum of two elements may be written as sup{x,y} or xy. 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.

TEdit

top 
See unit.
total order 
A partial order T in which, for each x and y in T, we have xy or yx. Total orders are also called linear orders or chains.
transitive 
Of a relation R on a set X, such that x R y and y R z imply x R z, for all elements x, y, z in X.

UEdit

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 xb, 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, xp implies that p is contained in X. The dual notion is called lower set.

WEdit

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, ysup D implies xd for some d in D. One also says that x approximates y. See also domain theory.

ZEdit

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.

ReferencesEdit

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

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