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A new approach to universal $F$-inverse monoids in enriched signatureGanna Kudryavtseva,
Ajda Lemut Furlani, 2024, original scientific article
Abstract: We show that the universal $X$-generated $F$-inverse monoid $F(G)$, where ▫$G$▫ is an $X$-generated group, introduced by Auinger, Szendrei and the first-named author, arises as a quotient inverse monoid of the Margolis-Meakin expansion $M(G, X\cup \overline{G})$ of $G$, with respect to the extended generating set $X\cup \overline{G}$, where $\overline{G}$ is a bijective copy of $G$ which encodes the ▫$m$▫-operation in $F(G)$. The construction relies on a certain dual-closure operator on the semilattice of all finite and connected subgraphs containing the origin of the Cayley graph ${\rm Cay}(G, X\cup {\overline{G}})$ and leads to a new and simpler proof of the universal property of $F(G)$.
Keywords: inverse monoid, F-inverse monoid, Margolis-Meakin expansion, group presentation, Cayley graph of a group, closure operator, dual-closure operator, partial action, partial action product
Published in DiRROS: 21.10.2024; Views: 730; Downloads: 358
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