Vítor Hugo Fernandes

In this paper we consider various classes of monoids of transformations of a finite chain,
including those of transformations that preserve or reverse either the order or the orientation.
In line with Howie and McFadden (1990),
we complete the study of the ranks (and of idempotent ranks, when applicable) of all their ideals.

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.

Summary: In this paper we consider the monoid $\mathcal {OR}_{n}$ of all full transformations on a chain with $n$ elements that preserve or reverse the orientation, as well as its submonoids $\mathcal {OD}_{n}$ of all order-preserving or order-reversing elements, $\mathcal {OP}_{n}$ of all orientation-preserving elements and $\mathcal {O}_{n}$ of all order-preserving elements. By making use of some well known presentations, we show that each of these four monoids is a quotient of a bilateral semidirect product of two of its remarkable submonoids.

A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup $TF_{n}$ of all
order-preserving transformations on an $n$-element zig-zag ordered set. We determine the rank of $TF_{n}$ and provide a minimal generating set for $TF_{n}$. Moreover, a formula for the number of idempotents in $TF_{n}$ is given.

In this paper we give presentations for the monoid $\DP_n$ of all partial isometries on $\{1,\ldots,n\}$ and for its submonoid $\ODP_n$ of all order-preserving partial isometries.

Following the new description of an oriented full transformation on a finite chain given recently by Higgins and Vernitsk,
in this short note we present a refinement of this description which is extendable to partial transformations and to injective partial transformations.

In this paper we consider the monoid DPSn of all partial isometries of a star graph Sn with n vertices. Our main objectives are to determine the rank and to exhibit a presentation of DPSn. We also describe Green’s relations of DPSn and calculate its cardinal.

In this paper we characterize the monoids of endomorphisms of the semigroups PO_n and POI_n of all order-preserving partial transformations and of all order-preserving partial permutations, respectively, of a finite n-chain.

In this note we consider the monoid $PODI_n$ of all monotone partial permutations on $\{1,\ldots,n\}$ and its submonoids $DP_n$, $POI_n$ and $ODP_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $POI_n$ and $ODP_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $PODI_n$ is a quotient of a semidirect product of $POI_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $DP_n$ is a quotient of a semidirect product of $ODP_n$ and $\mathcal{C}_2$.

In this paper we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid wEndP_n of all weak endomorphisms of a directed path with n vertices, which is a submonoid of the widely studied monoid O_n of all order-preserving transformations of a n-chain. Also, we describe the regular elements of wEndP_n and calculate its size and number of idempotents.

In this short note we construct an extension of the Vagner-Preston representation for block-groups and show that its kernel is the largest congruence that separates regular elements.