Tootfinder

Subfitness in distributive (semi)lattices
Guram Bezhanishvili, James Madden, M. Andrew Moshier, Marcus Tressl, Joanne Walters-Wayland
https://arxiv.org/abs/2404.06071

Subfitness in distributive (semi)lattices
We investigate whether the set of subfit elements of a distributive semilattice is an ideal. This question was raised by the second author at the BLAST conference in 2022. We show that in general it has a negative solution, however if the semilattice is a lattice, then the solution is positive. This is somewhat unexpected since, as we show, a semilattice is subfit if and only if so is its distributive lattice envelope.

On Finite Presentability of Subsemigroups of the Monogenic Free Inverse Semigroup
Yung Won Cho, Nik Ruskuc
https://arxiv.org/abs/2404.05244 https://…

On Finite Presentability of Subsemigroups of the Monogenic Free Inverse Semigroup
The monogenic free inverse semigroup $FI_1$ is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of $FI_1$ is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of $FI_1$ is finitely presented as a semigroup if and only if it is a finite semilattice.

This https://arxiv.org/abs/2402.14681 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Decomposition of Płonka sums into direct systems
The Płonka sum is an algebra determined using a structure called a direct system. By a direct system, we mean an indexed family of algebras with disjoint universes whose indexes form a join-semilattice s.t. if two indexes are in a partial order relation, then there is a homomorphism from the algebra of the first index to the algebra of the second index. The sum of the sets of the direct system determines the universe of the Płonka sum. Therefore, to speak about a Płonka sum, there must be a …

This https://arxiv.org/abs/2308.01627 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Semilattice base hierarchy for frames and its topological ramifications
We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice $A$, we analyze the interval in the coframe of sublocales of the frame of downsets of $A$ formed by all frames with the S-base $A$. We give an explicit description of the nuclei associated with these sublocales. We study various degrees of completeness of $A$, which generalize the concepts of extremally disconnected and basically disconnected frames. We also introduce the concepts of D-bas…

Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets
Ivan Chajda, Miroslav Kola\v{r}\'ik, Helmut L\"anger
https://arxiv.org/abs/2404.13361

Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets
On an arbitrary meet-semilattice S with 0 we define an orthogonality relation and investigate the lattice Cl(S) of all subsets of S closed under this orthogonality. We show that if S is atomic then Cl(S) is a complete atomic Boolean algebra. If S is a pseudocomplemented lattice, this orthogonality relation can be defined by means of the pseudocomplementation. Finally, we show that if S is a complete pseudocomplemented lattice then Cl(S) is a complete Boolean algebra. For pseudocomplemented pose…

This https://arxiv.org/abs/2308.01627 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Semilattice base hierarchy for frames and its topological ramifications
We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice $A$, we analyze the interval in the coframe of sublocales of the frame of downsets of $A$ formed by all frames with the S-base $A$. We give an explicit description of the nuclei associated with these sublocales. We study various degrees of completeness of $A$, which generalize the concepts of extremally disconnected and basically disconnected frames. We also introduce the concepts of D-bas…

Decomposition of P{\l}onka sums into direct systems
Krystyna Mruczek-Nasieniewska, Mateusz Klonowski
https://arxiv.org/abs/2402.14681 https://

Decomposition of Płonka sums into direct systems
The Płonka sum is an algebra determined using a structure called direct system. By a direct system, we mean an indexed family of algebras with disjoint universes whose indexes form a join-semilattice s.t. if two indexes are in a partial order relation, then there is a homomorphism from the algebra of the first index to the algebra of the second index. The sum of the sets of the direct system determines the universe of Płonka sum. Therefore, to speak about a Płonka sum, there must be a direct…

This https://arxiv.org/abs/2402.12313 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A new approach to universal $F$-inverse monoids in enriched signature
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 closure operator on the semilattice …

This https://arxiv.org/abs/2402.14681 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Decomposition of Płonka sums into direct systems
The Płonka sum is an algebra determined using a structure called a direct system. By a direct system, we mean an indexed family of algebras with disjoint universes whose indexes form a join-semilattice s.t. if two indexes are in a partial order relation, then there is a homomorphism from the algebra of the first index to the algebra of the second index. The sum of the sets of the direct system determines the universe of the Płonka sum. Therefore, to speak about a Płonka sum, there must be a …

This https://arxiv.org/abs/2403.05514 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Convex Geometry of Building Sets
Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust combinatorial abstraction of convexity. Supersolvable convex geometries and antimatroids make appearances in the study of poset closure operators, Coxeter groups, and matroid activities. We prove that the building sets on a meet-semilattice form a supersolvable convex geometry.

Inverse semigroups of separated graphs and associated algebras
Pere Ara, Alcides Buss, Ado Dalla Costa
https://arxiv.org/abs/2403.05295 https://

Inverse semigroups of separated graphs and associated algebras
In this paper we introduce an inverse semigroup $\mathcal{S}(E,C)$ associated to a separated graph $(E,C)$ and describe its internal structure. In particular we show that it is strongly $E^*$-unitary and can be realized as a partial semidirect product of the form $\mathcal{Y}\rtimes\mathbb{F}$ for a certain partial action of the free group $\mathbb{F}=\mathbb{F}(E^1)$ on the edges of $E$ on a semilattice $\mathcal{Y}$ realizing the idempotents of $\mathcal{S}(E,C)$. In addition we also describe…

This https://arxiv.org/abs/2402.14681 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Decomposition of Płonka sums into direct systems
The Płonka sum is an algebra determined using a structure called a direct system. By a direct system, we mean an indexed family of algebras with disjoint universes whose indexes form a join-semilattice s.t. if two indexes are in a partial order relation, then there is a homomorphism from the algebra of the first index to the algebra of the second index. The sum of the sets of the direct system determines the universe of the Płonka sum. Therefore, to speak about a Płonka sum, there must be a …

A new approach to universal $F$-inverse monoids in enriched signature
Ganna Kudryavtseva, Ajda Lemut
https://arxiv.org/abs/2402.12313 https://

A new approach to universal $F$-inverse monoids in enriched signature
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 closure operator on the semilattice …

Convex Geometry of Building Sets
Spencer Backman, Rick Danner
https://arxiv.org/abs/2403.05514 https://arxiv.org/pdf/2403.05514

Convex Geometry of Building Sets
Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust combinatorial abstraction of convexity. Supersolvable convex geometries and antimatroids make appearances in the study of poset closure operators, Coxeter groups, and matroid activities. We prove that the building sets on a meet-semilattice form a supersolvable convex geometry.

Semiring arising as Lattice of Groupsemirings
A. R. Rajan, S. Sheena, C. S. Preenu
https://arxiv.org/abs/2402.10103 https://arxiv.org…

Semiring arising as Lattice of Groupsemirings
Much study has been done on semigroups which are unions of groups. There are several ways in which a union of groups can be made into a semigroup in which each of the component groups arises as subgroups of the constructed semigroup. An important class of such unions is a semilattice of groups. Group semirings are semirings $(G,+,\cdot )$ where $(G,\cdot )$ is a group and $(G,+)$ is a left zero semigroup. We consider construction of semirings from classes of group semirings $\{G_α:α\in D \}$ …