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- if x ≤ y, then C(x) ≤ C(y), i.e. C is montonically increasing
- x ≤ C(x) for all x
- C(C(x)) = C(x) for all x, i.e. C is an idempotent function
The name comes from the fact that forming the closure of subsets of a topological space has these properties if the set of all subsets is ordered by inclusion ⊆. (Note that the topological closure operator is not characterized by these properties however; see the Kuratowski closure axioms for a complete characterization.)
Suppose you have some logical formalism that contains certain rules allowing you to derive new formulas from given ones. Consider the set F of all possible formulas, and let P be the power set of F, ordered by ⊆. For a set X of formulas, let C(X) be the set of all formulas that can be derived from X. Then C is a closure operator on P.
Another typical closure operator is the following: take a group G and for any subset X of G, let C(X) be the subgroup generated by X, i.e. the smallest subgroup of G containing X. Then C is a closure operator on the set of subsets of G, ordered by inclusion ⊆. Analogous examples can be given for the subspace generated by a given subset of a vector space, for the subfield generated by a given subset of a field, or indeed for the subalgebra generated by a given subset of any algebra in the sense of universal algebra.
Closed elements; properties
Given a closure operator C, a closed element of P is an element x that is a fixed point of C, or equivalently, that is in the image of C. If a is closed and x is arbitrary, then we have x ≤ a if and only if C(x) ≤ a. So C(x) is the smallest closed element that's greater than or equal to x. We see that C is uniquely determined by the set of closed elements.
Every Galois connection gives rise to a closure operator (as is explained in that article). In fact, every closure operator arises in this way from a suitable Galois connection. The Galois connection is not uniquely determined by the closure operator. One Galois connection that gives rise to the closure operator C can be described as follows: if A is the set of closed elements with respect to C, then C : P → A is the lower adjoint of a Galois connection between P and A, with the upper adjoint being the embedding of A into P. Furthermore, every lower adjoint of an embedding of some subset into P is a closure operator. "Closure operators are lower adjoints of embeddings." Note however that not every embedding has a lower adjoint.
Any partially ordered set P can be viewed as a category, with a single morphism from x to y if and only if x ≤ y. The closure operators on the partially ordered set P are then nothing but the monads on the category P.
If P is a complete lattice, then a subset A of P is the set of closed elements for some closure operator on P if and only if A is a Moore family on P, i.e. the largest element of P is in A, and the infimum (meet) of any non-empty subset of A is again in A. Any such set A is itself a complete lattice with the order inherited from P (but the supremum (join) operation might differ from that of P). The closure operators on P form themselves a complete lattice; the order on closure operators is defined by C1 ≤ C2 iff C1(x) ≤ C2(x) for all x in P.
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