Moore closure

Moore closures


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.


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


Let XX be a set, and let π’žβŠ‚π’«X\mathcal{C} \subset \mathcal{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} .

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}.


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


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 isotone, extensive, and idempotent. That is,

  1. AβŠ†Bβ‡’Cl(A)βŠ†Cl(B) A \subseteq B \;\Rightarrow\; Cl(A) \subseteq Cl(B) ,
  2. AβŠ†Cl(A) A \subseteq Cl(A) , and
  3. Cl(Cl(A))βŠ†Cl(A) Cl(Cl(A)) \subseteq Cl(A) (the reverse inclusion follows from the previous two properties).

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 sets 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. , corresponds to 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 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):

(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:

But also:

Here are some examples on power sets:

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.


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:

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.


Section 4.1–4.12 in

See also