Abstract
We show that the category of X-generated E-unitary inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of connected subgraphs of the Cayley graph of G. Analogously, we study F-inverse monoids in the extended signature \((\cdot, 1, ^{-1}, ^\mathfrak m)\), and show that the category of X-generated F-inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of all subgraphs of the Cayley graph of G. As an application, we show that presentations of F-inverse monoids in the extended signature can be studied by tools analogous to Stephen’s procedure in inverse monoids, in particular, we introduce the notions of F-Schützenberger graphs and P-expansions.
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Acknowledgement
The author would like to thank Mark Kambites and Benjamin Steinberg for helpful conversations.
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Dedicated to Mária B. Szendrei on the occasion of her 70th birthday
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Szakács, N. E-unitary and F-inverse monoids, and closure operators on group Cayley graphs. Acta Math. Hungar. 173, 297–316 (2024). https://doi.org/10.1007/s10474-024-01443-w
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DOI: https://doi.org/10.1007/s10474-024-01443-w