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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

  1. $\mathbf c(0) = 0'$,
  2. For all $a \in L$, $a \leq \mathbf c(a)$,
  3. For all $a \in L$, $\mathbf c^2(A) = \mathbf c(A)$,
  4. 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.

giobrach
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  • Your third point doesn't make sense. If $\mathbf{c}$ is a map from $L$ to $L'$, then you cannot apply it twice. – Arturo Magidin Aug 21 '19 at 17:40
  • @ArturoMagidin, $L'$ is required to be a subset of $L$. – giobrach Aug 21 '19 at 17:41
  • And you aren't just talking about a Galois connection when you consider both $\mathbf{c}$ and the inclusion map? I guess "axiom 1" is a bit of an obstacle, but I would argue that the axiom isn't really that useful as written, as you can just declare $\mathbf{0}'$ to be whatever $\mathbf{c}(\mathbf{0})$ is. – Arturo Magidin Aug 21 '19 at 17:42
  • Thanks. It does seem like these generalized closures are instances of something like a Galois connection between the map "smallest substructure containing" and simple inclusion. However, my elaboration leads to something more specific, due to the fact that I want to include axiom 4, and thus I need a lattice structure. Besides the issue with axiom 1, it looks like what you say is exactly what I need provided $\mathbf c$ is only a Moore closure. – giobrach Aug 21 '19 at 18:04

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