Closure relation

closure operation, in a partially ordered set $M$

A single-valued mapping of $M$ into itself, associating with each element $a \in M$ an element $\bar a \in M$, called the closure of $a$, in such a way that the following conditions hold: 1) $a \le \bar a$; 2) if $a \le b$, then $\bar a \le \bar b$; and 3) $\overline{\bar a} = \bar a$. An element $a \in M$ is said to be closed if it is its own closure. A closure operation in a set $M$ is uniquely determined by specifying the system of all closed elements.

In the particular case when $M$ is the set of all subsets of an arbitrary set $X$, ordered by inclusion, one speaks of a closure operation on $X$. On any set $X$, a closure operation can be defined by taking the closed subsets to be any system of subsets that includes $X$ itself and is closed under arbitrary intersections. Two partially ordered sets with closure operations are said to be isomorphic if there is an isomorphism of the partially ordered sets under which the images and pre-images of closed sets are closed.

Considerable importance is given in mathematics to closure operations on the set of all subsets of $X$ which satisfy the following additional assumption: The empty set is closed, and the closure of the union of two subsets of $X$ equals the union of their closures. A closure operation satisfying this condition is called a topology on the set $X$: cf. Topological structure (topology)

References

- P.M. Cohn, "Universal algebra", Reidel (1981) MR0620952 Zbl 0461.08001

- A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian) MR0158000 Zbl 0121.25901

Comments

Instead of closure operation the terms closure operator and join operator are also used (cf., e.g., [1]). A family of subsets of a given set $X$ is called a closure system if it is closed under arbitrary intersections. An example is the collection of all subspaces of an affine space $A$, and as explained above this induces a closure operation on the subsets of $A$.

Closure operations commuting with finite unions are often called Kuratowski closure operators, in honour of [a1]. A Boolean algebra equipped with a closure operation is sometimes called a closure algebra (see [a2]).

References

[a1]	C. Kuratowski, "Sur l'opération $\bar A$ de l'analysis situs" Fund. Math. , 3 (1922) pp. 182–199
[a2]	J.C.C. McKinsey, A. Tarski, "On closed elements in closure algebras" Ann. Math. (2) , 47 (1946) pp. 122–162