Are there references on a possible generalization of Kuratowski closure operators, namely maps $\mathbf c: L \to L'$, where $(L,\wedge,\vee,\leq,0,1)$ and $(L',\wedge',\vee',\leq',0',1')$ are bounded lattices with $L' \subseteq L$, such that
- $\mathbf c(0) = 0'$,
- For all $a \in L$, $a \leq \mathbf c(a)$,
- For all $a \in L$, $\mathbf c^2(A) = \mathbf c(A)$,
- For all $a,b \in L$, $\mathbf c(a \vee b) = \mathbf c(a) \vee' \mathbf c(b)$?
The usual Kuratowski closure operator is retrieved when all primed structures coincide with the corresponding non-primed structure.
There seem to be many examples of this kind of structure:
- If $V$ is a vector space, we may take $L = \wp^2(V)$, the collection of all families of subsets of $V$, and the usual lattice structure generated by inclusion; then choose $L' = \mathrm{Ssp}(V)$, the collection of all subspaces of $V$, with $\wedge' = \cap$, $\vee' = \oplus$, $0' = \{0\}$, $1 = V$. Define $\mathbf c : \wp^2(V) \to \mathrm{Ssp}(V)$ such that $\mathbf c(\mathscr F)$ is the subspace generated by the elements of $\mathscr F$.
- Similarly for groups, rings, fields, just with subgroups (or normal subgroups), subrings (or ideals), subfields in place of vector subspaces.
- If $X$ is a nonempty set, we may take $L = \wp^2(X)$ as above, and $L'=\mathrm{Top}(X)$, the collection of all topologies on $X$, with $\wedge' = \cap$, $\vee'$ such that $\tau_1 \vee' \tau_2$ is the smallest topology containing $\tau_1 \cup \tau_2$. Define $\mathbf c$ such that $\mathbf c(\mathscr F)$ is the topology generated by $\mathscr F$ as a subbasis.
- Similarly for the family of all $\sigma$-algebras, or of filters, on a nonempty set $X$...
Note. This idea is akin to, but ultimately distinct from, that of Moore (algebraic) closure operators (cfr. Burris & Sankappanavar, A Course in Universal Algebra, p. 20ff.). Moore closures are not required to satisfy the analogous to axiom 1 (preservation of the bottom element) and are only required to be isotonic, so a lattice structure is not necessary as in the case of Kuratowski operators, just a partial order.