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Vítor H. Fernandes

Publications and source records attributed to Vítor H. Fernandes.

At least 19 recordsLinked to original sources
arXiv2026

Groups of permutations that are even on maximal proper subsets, and related monoids

Let $n$ be a positive integer and let $[n]=\{1,2,\ldots,n\}$. Let $Γ_n$ denote the group of permutations on $[n]$ whose restrictions to maximal proper subsets of $[n]$ are even, let $Σ_n$ denote the monoid of transformations on $[n]$ whose injective restrictions to maximal proper subsets of $[n]$ are even and let $Δ_n$ denote the submonoid of $Σ_n$ generated by transformations of rank at least $n-1$. In this paper, we present descriptions of $Γ_n$, $Δ_n$ and $Σ_n$, determine their cardinalities and ranks, and provide minimal generating sets for each of them.

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arXiv2025

On monoids of monotone partial transformations of a finite chain whose domains and ranges are intervals

In this note, we consider the monoid $\mathcal{PIM}_{n}$ of all partial monotone transformations on a chain with $n$ elements whose domains and ranges are intervals and its submonoid $\mathcal{IM}_{n}$ constituted by the full transformations. For both of these monoids, our aim is to determine their cardinalities and ranks and define them by means of presentations. We also calculate the number of nilpotent elements of $\mathcal{PIM}_{n}$.

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arXiv2025

On the monoid of partial order-preserving transformations of a finite chain whose domains and ranges are intervals

In this paper, we consider the monoid $\mathcal{PIO}_{n}$, of all partial order-preserving transformations on a chain with $n$ elements whose domains and ranges are intervals, along with its submonoid $\mathcal{PIO}_{n}^-$ of order-decreasing transformations. Our main aim is to give presentations for $\mathcal{PIO}_{n}^-$ and $\mathcal{PIO}_{n}$. Moreover, for both monoids, we describe regular elements and determine their ranks, cardinalities and the numbers of idempotents and nilpotents.

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arXiv2024

Presentations for monoids of partial endomorphisms of a star graph

In this paper, we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

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arXiv2024

Oriented transformations on a finite chain: another description

Following the new description of an oriented full transformation on a finite chain given recently by Higgins and Vernitski in "Orientation-preserving and orientation-reversing mappings: a new description", Semigroup Forum 104 (2022), 509--514, in this short note we present a refinement of this description which is extendable to partial transformations and to injective partial transformations.

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arXiv2024

On partial endomorphisms of a star graph

In this paper we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to determine their ranks. We also describe their Green's relations, calculate their cardinalities and study their regularity.

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arXiv2024

On semigroups of orientation-preserving partial permutations with restricted range

Let $Ω_n$ be a finite chain with $n$ elements $(n\in\mathbb{N})$, and let $\mathcal{POPI}_{n}$ be the semigroup of all injective orientation-preserving partial transformations of $Ω_n$. In this paper, for any nonempty subset $Y$ of $Ω_n$, we consider the subsemigroup $\mathcal{POPI}_{n}(Y)$ of $\mathcal{POPI}_{n}$ of all transformations with range contained in $Y$. We describe the Green's relations and study the regularity of $\mathcal{POPI}_{n}(Y)$. Moreover, we calculate the rank of $\mathcal{POPI}_{n}(Y)$ and determine when two semigroups of this type are isomorphic.

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arXiv2023

On three submonoids of the dihedral inverse monoid on a finite set

In this paper we consider three submonoids of the dihedral inverse monoid $\mathcal{DI}_n$, namely its submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green's relations and determine the rank.

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arXiv2023

On monoids of endomorphisms of a cycle graph

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities.

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arXiv2023

On the monoid of partial isometries of a wheel graph

In this paper, we consider the monoid $DPW_n$ of all partial isometries of a wheel graph $W_n$ with $n+1$ vertices. Our main objective is to determine the rank of $DPW_n$. In the process, we also compute the ranks of three notable subsemigroups of $DPW_n$. We also describe Green's relations of $DPW_n$ and of its three considered subsemigroups.

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arXiv2022

On the monoid of partial isometries of a cycle graph

In this paper we consider the monoid $\DPC_n$ of all partial isometries of a $n$-cycle graph $C_n$. We show that $\DPC_n$ is the submonoid of the monoid of all oriented partial permutations on a $n$-chain whose elements are precisely all restrictions of the dihedral group of order $2n$. Our main aim is to exhibit a presentation of $\DPC_n$. We also describe Green's relations of $\DPC_n$ and calculate its cardinal and rank.

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arXiv2022

Endomorphisms of semigroups of oriented transformations

In this paper, we characterize the monoid of endomorphisms of the semigroup of all oriented full transformations of a finite chain, as well as the monoid of endomorphisms of the semigroup of all oriented partial transformations and the monoid of endomorphisms of the semigroup of all oriented partial permutations of a finite chain. Characterizations of the monoids of endomorphisms of the subsemigroups of all orientation-preserving transformations of the three semigroups aforementioned are also given. In addition, we compute the number of endomorphisms of each of these six semigroups.

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arXiv2022

Endomorphisms of semigroups of monotone transformations

In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

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arXiv2021

Partial Automorphisms and Injective Partial Endomorphisms of a Finite Undirected Path

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids $IEnd(P_n)$ and $PAut(P_n)$ of all injective partial endomorphisms and of all partial automorphisms of the undirected path $P_n$ with $n$ vertices. We also describe Green's relations of $PAut(P_n)$ and $IEnd(P_n)$ and calculate their cardinals.

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arXiv2019

Ranks and presentations of some normally ordered inverse semigroups

In this paper we compute the rank and exhibit a presentation for the monoids of all $P$-stable and $P$-order preserving partial permutations on a finite set $Ω$, with $P$ an ordered uniform partition of $Ω$. These (inverse) semigroups constitute a natural class of generators of the pseudovariety of inverse semigroups ${\sf NO}$ of all normally ordered (finite) inverse semigroups.

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