1. $F$-birestriction monoids in enriched signatureGanna Kudryavtseva, Ajda Lemut Furlani, 2025, izvirni znanstveni članek Povzetek: Motivated by recent interest to $F$-inverse monoids, on the one hand, and to restriction and birestriction monoids, on the other hand, we initiate the study of $F$-birestriction monoids as algebraic structures in the enriched signature $(\cdot, \, ^*, \,^+, \, ^\mathfrak{m},1)$ where the unary operation $(\cdot)^\mathfrak{m}$ maps each element to the maximum element of its $\sigma$-class. We find a presentation of the free $F$-birestriction monoid ${\mathsf{FFBR}}(X)$ as a birestriction monoid ${\mathcal F}$ over the extended set of generators $X\cup\overline{X^+}$ where $\overline{X^+}$ is a set in a bijection with the free semigroup $X^+$ and encodes the maximum elements of (non-projection) $\sigma$-classes. This enables us to show that ${\mathsf{FFBR}}(X)$ decomposes as the partial action product $E({\mathcal I})\rtimes X^*$ of the idempotent semilattice of the universal inverse monoid ${\mathcal I}$ of ${\mathcal F}$ partially acted upon by the free monoid $X^*$. Invoking Schützenberger graphs, we prove that the word problem for ${\mathsf{FFBR}}(X)$ and its strong and perfect analogues is decidable. Furthermore, we show that ${\mathsf{FFBR}}(X)$ does not admit a geometric model based on a quotient of the Margolis-Meakin expansion $M({\mathsf{FG}}(X), X\cup \overline{X^+})$ over the free group ${\mathsf{FG}}(X)$, but the free perfect $X$-generated $F$-birestriction monoid admits such a model.
Ključne besede: birestriction monoid, F-birestriction monoid, free F-birestriction monoid, inverse monoid, F-inverse monoid, Margolis-Meakin expansion, Schützenberger graph, partial action, partial action product
Objavljeno v DiRROS: 03.11.2025; Ogledov: 154; Prenosov: 80
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2. Monoid algebras and graph productsWilfried Imrich, Igor Klep, Daniel Smertnig, 2025, izvirni znanstveni članek Povzetek: In this note, we extend results about unique $n^{\rm th}$ roots and cancellation of finite disconnected graphs with respect to the Cartesian, the strong and the direct product, to the rooted hierarchical products, and to a modified lexicographic product. We show that these results also hold for graphs with countably many finite connected components, as long as every connected component appears only finitely often (up to isomorphism). The proofs are via monoid algebras and generalized power series rings.
Ključne besede: graph products, monoid algebras, power series rings, uniqueness of roots, cancellation property
Objavljeno v DiRROS: 29.04.2025; Ogledov: 561; Prenosov: 237
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3. A new approach to universal $F$-inverse monoids in enriched signatureGanna Kudryavtseva, Ajda Lemut Furlani, 2024, izvirni znanstveni članek Povzetek: 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 $F(G)$.
Ključne besede: 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
Objavljeno v DiRROS: 21.10.2024; Ogledov: 621; Prenosov: 315
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4. Globalization of partial actions of semigroupsGanna Kudryavtseva, Valdis Laan, 2023, izvirni znanstveni članek Povzetek: We propose two universal constructions of globalization of a partial action of a semigroup on a set, satisfying certain conditions which arise in Morita theory of semigroups. One of the constructions is based on the tensor product of a partial semigroup act with the semigroup and generalizes the globalization construction of strong partial actions of monoids due to Megrelishvili and Schröder. It produces the initial object in an appropriate category of globalizations of a given partial action. The other construction involves Hom-sets and is novel even in the monoid setting. It produces the terminal object in an appropriate category of globalizations. While in the group case the results of the two constructions are isomorphic, they can be far different in the monoid case.
Ključne besede: partial action, partial semigroup action, partial monoid action, globalization, enveloping action
Objavljeno v DiRROS: 18.03.2024; Ogledov: 1140; Prenosov: 529
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