\(E\)-unitary and \(F\)-inverse monoids, and closure operators on group Cayley graphs (Q6607776)

An algebra \(S\) is called \(X\)-generated via the map \(\iota: X \to S\) if \(S\) is generated by \(\iota (X)\). Let \(S_1,S_2\) be two \(X\)-generated algebras via the maps \(\iota_1: X \to S_1\) and \(\iota_2: X \to S_2\). A homomorphism \(\psi: S_1 \to S_2\) is called canonical, if \(\iota_2 = \psi \circ \iota_1\). For an inverse semigroup \(M\), let \(\sigma\) denote the minimum group congruence of \(M\). An inverse semigroup \(M\) is called E-unitary if every element greater than an idempotent is idempotent as well and \(M\) is called F-unitary if each \(\sigma\)-class has a greatest element (with respect to the natural partial order on \(M\)).\par The categories \(\mathcal{E}(X,G)\) and \(\mathcal{F}(X,G)\) of \(X\)-generated E-unitary and respectively F-unitary inverse monoids \(M\) are considered where \(G\) is greatest group image of \(M\) considered up to canonical isomorphism, and the morphisms are canonical inverse monoid homomorphisms.\par For an \(X\)-generated inverse monoid \(S\) (in particular, a group), the Cayley graph \(\Gamma_X\) of \(S\) is defined. Let Sub\(\Gamma_X\) and CSub\( \Gamma_X\) be the set of all subgraphs and the set of all connected subgraphs of \(\Gamma_X\) respectively. The main theorem of the paper says that for an X-generated group G (1) the category \(\mathcal{E}(X,G)\) is equivalent to the category \(C(X,G)\) of \(G\)-invariant, finitary closure operators on (CSub\(\Gamma_X,\subseteq)\) and (2) the category \(\mathcal{F}(X,G)\) is equivalent to the category \(S(X,G)\) of \(G\)-invariant, finitary closure operators on (Sub\(\Gamma_X,\subseteq)\).