The concept of Moore closure is a very general idea of what it can mean for a set to be closed under some condition. It includes, as special cases, the operation of closure in a topological space, many examples of generation of structures from bases and even subbases, and generating subalgebras from subsets of an algebra.
Secretly, it is the same thing as the collection of subsets preserved by some monad on a power set (the subset of “modal types”). In fact it is a special case of the notion of closure operator or modality in logic/type theory, namely the special case where the ambient category/hyperdoctrine is the topos Set.
We give two equivalent definitions. The first one
on the subobject lattice of the given set.
That is, given a family of sets in ,
Given any collection whatsoever of subsets of , the Moore collection generated by is the collection of all intersections of members of .
This is indeed a Moore collection, and it equals if and only if is a Moore collection.
Again let be a set, and now let be an operation on subsets of . Then is a closure operation if is monotone, isotone, and idempotent. That is,
Conversely, if is a Moore collection, then let be the intersection of all closed sets that contain . Then is a closure operator.
Furthermore, the two maps above, from closure operators to Moore collections and vice versa, are inverses.
Moore closures on are precisely monads on the subobject lattice . The property (1) of a closure operator, def. 3 ,corresponds the action of the monad on morphisms, while (2,3) are the unit and multiplication of the monad. (The rest of the requirements of a monad are trivial in a poset, since they state the equality of various morphisms with common source and target.)
What are examples? Better to ask what isn't an example! (Answer: preclosure in a pretopological space, even though some authors call this ‘closure’.)
Of course, the closed subsets in a topological space form a Moore collection; then the closure of a set is its closure in the usual sense. In fact, a topological space can be defined as a set equipped with a Moore closure with either of these additional properties (which are equivalent):
(However, these properties may fail in constructive mathematics; in fact, a topology cannot be constructively recovered from its closure operation.)
Here are some algebraic examples: * The subgroups of a group form a Moore collection; the closure of a subset of is the subgroup generated by . * The subrings of a ring form a Moore collection; the closure of a subset of is the subring generated by . * The subspaces of a vector space form a Moore collection; the closure of a subset of is the subspace spanned by . * OK, you get the idea. This applies to any algebraic theory. For a finitary algebraic theory, the lattice of closed elements is an algebraic lattice.
But also: * The normal subgroups of form a Moore collection; the closure of is the normal subgroup generated by . * The ideals of a ring form a Moore collection; the closure of is the ideal generated by . * The (topologically) closed subspaces of a Hilbert space form a Moore collection; the closure of is the closed subspace generated by . * And many further examples.
Here are some examples on power sets: * The topologies on form a Moore collection on ; the closure of a subset of is the topology generated by as a subbase. * The filters on form a Moore collection on ; the closure of is the filter generated by as a subbase. (The proper filters on do not form a Moore collection; not every generates a proper filter.) * The -algebras on form a Moore collection on ; the closure of is the -algebra generated by . (This is the ‘abstract nonsense’ way to generate a -algebra; else you have to do transfinite induction on countable ordinals.) * And so on.
Topping off these, the Moore collections on form a Moore collection on ; the closure of is the Moore collection generated by as described in the definitions.
See also at matroid.
The definition of Moore collection really makes sense in any inflattice; even better, the definition of closure operator makes sense in any poset. This context is the generic meaning of closure operator; here are some examples:
Instead of , work in the opposite poset . Then the open sets in a topological space form a Moore collection whose closure operator is the usual interior operation. Now we can define a topological space as a set equipped with a Moore closure operator on that preserves joins (which here are intersections); this definition is even valid constructively.
Let be a Galois connection between posets and . Call an element of normal if (the reverse is always true). Then is a closure operator. This generalises the case of the normal subgroups of when is the Galois group of an extension of fields.
Section 4.1–4.12 in