Tue, 14 Nov 2023, 1:25 pm MT
Inverse monoids are an important generalization of groups, and they form a variety of algebras. We focus on the class of inverse monoids called F-inverse: this is not closed under taking inverse subsemigroups or homomorphic images, but does form a variety in the signature enriched with an extra unary operation. Recently, Auinger, Kudryavtseva and M. Szendrei gave an elegant model for free F-inverse monoids in this extended signature. More generally, they describe the initial object in the category of X-generated F-inverse monoids with greatest group image G. We show that this category is equivalent to the category of finitary (i.e. algebraic) G-invariant closure operators on subgraphs of the Cayley graph of G, and adapt Stephen's approach to study the word problem in presentations of inverse monoids for F-inverse monoids in the enriched signature.
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