nLab
Moore closure

Moore closures

Idea

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 β€œmodal 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.

Definitions

We give two equivalent definitions. The first one

gives the explicit condition for a subset of a power set to qualify as a Moore closure, the second

characterizes Moore closures as the collections of modal types of suitable closure operators. More abstractly, this characterizes Moore closures

on the subobject lattice of the given set.

In terms of closure condition

Definition

Let XX be a set, and let π’žβŠ‚PX\mathcal{C} \subset P X be a collection of subsets of XX. Then π’ž\mathcal{C} is a Moore collection if every intersection of members of π’ž\mathcal{C} belongs to π’ž\mathcal{C}.

That is, given a family (A i) i(A_i)_i of sets in XX,

βˆ€i,A iβˆˆπ’žβ‡’β‹‚ iA iβˆˆπ’ž. \forall i,\; A_i \in \mathcal{C} \;\Rightarrow\; \bigcap_i A_i \in \mathcal{C} .
Definition

Given any collection ℬ\mathcal{B} whatsoever of subsets of XX, the Moore collection generated by ℬ\mathcal{B} is the collection of all intersections of members of ℬ\mathcal{B}.

Remark

This is indeed a Moore collection, and it equals ℬ\mathcal{B} if and only if ℬ\mathcal{B} is a Moore collection.

In terms of closure operators

Definition

Again let XX be a set, and now let ClCl be an operation on subsets of XX. Then ClCl is a closure operation if ClCl is monotone, isotone, and idempotent. That is, 1. AβŠ†Bβ‡’Cl(A)βŠ†Cl(B) A \subseteq B \;\Rightarrow\; Cl(A) \subseteq Cl(B) , 1. AβŠ†Cl(A) A \subseteq Cl(A) , and 1. Cl(Cl(A))βŠ†Cl(A) Cl(Cl(A)) \subseteq Cl(A) (the reverse inclusion follows from the previous two properties).

Proposition

If ClCl is a closure operation, then let π’ž\mathcal{C} be the collection of sets that equal their own closures (the β€œmodal types” or β€œlocal objects”). Then π’ž\mathcal{C} is a Moore collection.

Conversely, if π’ž\mathcal{C} is a Moore collection, then let Cl(A)Cl(A) be the intersection of all closed ssets that contain AA. Then ClCl is a closure operator.

Furthermore, the two maps above, from closure operators to Moore collections and vice versa, are inverses.

In terms of monads

Moore closures on XX are precisely monads on the subobject lattice 𝒫X\mathcal{P}X. 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.)

Examples

What are examples? Better to ask what isn't an example! (Answer: preclosure in a pretopological space, even though some authors call this β€˜closure’.)

Of course, the closed subsets in a topological space form a Moore collection; then the closure of a set AA 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):

  • Cl(βˆ…)=βˆ…Cl(\empty) = \empty and Cl(AβˆͺB)=Cl(A)βˆͺCl(B)Cl(A \cup B) = Cl(A) \cup Cl(B).
  • βˆ…\empty is closed, and so is AβˆͺBA \cup B if AA and BB are closed.

(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 GG form a Moore collection; the closure of a subset BB of GG is the subgroup generated by BB. * The subrings of a ring RR form a Moore collection; the closure of a subset BB of RR is the subring generated by BB. * The subspaces of a vector space VV form a Moore collection; the closure of a subset BB of VV is the subspace spanned by BB. * 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 GG form a Moore collection; the closure of BB is the normal subgroup generated by BB. * The ideals of a ring RR form a Moore collection; the closure of BB is the ideal generated by BB. * The (topologically) closed subspaces of a Hilbert space HH form a Moore collection; the closure of BB is the closed subspace generated by BB. * And many further examples.

Here are some examples on power sets: * The topologies on XX form a Moore collection on 𝒫X\mathcal{P}X; the closure of a subset ℬ\mathcal{B} of 𝒫X\mathcal{P}X is the topology generated by ℬ\mathcal{B} as a subbase. * The filters on XX form a Moore collection on 𝒫X\mathcal{P}X; the closure of ℬ\mathcal{B} is the filter generated by ℬ\mathcal{B} as a subbase. (The proper filters on XX do not form a Moore collection; not every ℬ\mathcal{B} generates a proper filter.) * The Οƒ\sigma-algebras on XX form a Moore collection on 𝒫X\mathcal{P}X; the closure of ℬ\mathcal{B} is the Οƒ\sigma-algebra generated by ℬ\mathcal{B}. (This is the β€˜abstract nonsense’ way to generate a Οƒ\sigma-algebra; else you have to do transfinite induction on countable ordinals.) * And so on.

Topping off these, the Moore collections on XX form a Moore collection on 𝒫X\mathcal{P}X; the closure of ℬ\mathcal{B} is the Moore collection generated by ℬ\mathcal{B} as described in the definitions.

See also at matroid.

Generalisations

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 𝒫X\mathcal{P}X, work in the opposite poset 𝒫 opX\mathcal{P}^{op}X. Then the open sets in a topological space XX 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 𝒫 opX\mathcal{P}^{op}X that preserves joins (which here are intersections); this definition is even valid constructively.

  • Let f⊣gf \dashv g be a Galois connection between posets AA and BB. Call an element of AA normal if g(f(a))≀ag(f(a)) \leq a (the reverse is always true). Then g∘fg \circ f is a closure operator. This generalises the case of the normal subgroups of GG when GG is the Galois group of an extension of fields.

Since Galois connections are simply adjunctions between posets, the concept of Moore closure cries out for categorification. And in fact, the answer is well known in category theory: it is a monad.

References

Section 4.1–4.12 in

See also

Revised on February 13, 2014 08:51:52 by Todd Trimble (67.81.95.215)