Vítor Hugo Fernandes

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.

In this paper, we consider the inverse submonoids AM_n of monotone transformations and AO_n of order-preserving transformations of the alternating inverse monoid AI_n on a chain with n elements. We compute the cardinalities, describe the Green's structures and the congruences, and calculate the ranks of these two submonoids of AI_n.

In this note we give a presentation for the monoid IO_n of all order-preserving transformations of a n-chain whose ranges are intervals. We also consider the submonoid IO_n^- of IO_n consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.

In this paper we study the cyclic inverse monoid CI_n on a set Ω_n with n elements, i.e. the inverse submonoid of the symmetric inverse monoid on Ω_n consisting of all restrictions of the elements of a cyclic subgroup of order n acting cyclically on Ω_n. We show that CI_n has rank 2 (for n⩾2) and n⋅2^n−n+1 elements. Moreover, we give presentations of CI_n on n+1 generators and (n^2+3n+4)/2 relations and on 2 generators and (n^2−n+6)/2 relations. We also consider the remarkable inverse submonoid OCI_n of CI_n constituted by all its order-preserving transformations. We show that OCI_n has rank n and 3⋅2^n−2n−1 elements. Furthermore, we exhibit presentations of OCI_n on n+2 generators and (n^2+3n+8)/2 relations and on n generators and (n^2+3n)/2 relations.

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 paper we introduce the notion of an orientation-preserving transformation on an arbitrary chain, as
a natural extension for infinite chains of the well known concept for finite chains introduced in 1998 by McAlister and, independently, in 1999 by Catarino and Higgins.
We consider the monoid POP(X) of all orientation-preserving partial transformations on a finite or infinite chain X and its submonoids OP(X) and POPI(X) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on X, respectively.
The monoid PO(X) of all order-preserving partial transformations on X and its injective counterpart POI(X) are also considered.
We study the regularity and give descriptions of the Green's relations of the monoids POP(X), PO(X), OP(X), POPI(X) and POI(X).

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 the inverse submonoids AOR_n of oriented transformations and AOP_n of orientation-preserving transformations of the alternating inverse monoid AI_n on a chain with n elements. We compute the cardinalities, describe the Green's structures and the congruences, and calculate the ranks of AOR_n and AOP_n.

Let $X$ be a finite or infinite chain and let $\O(X)$ be the monoid of all endomorphisms of $X$.
In this paper, we describe the largest regular subsemigroup of $\O(X)$ and Green's relations on $\O(X)$.
In fact, more generally, if $Y$ is a nonempty subset of $X$ and $\O(X,Y)$ is the subsemigroup of $\O(X)$ of all elements with range contained in $Y$,
we characterize the largest regular subsemigroup of $\O(X,Y)$ and Green's relations on $\O(X,Y)$.
Moreover, for finite chains, we determine when two semigroups of the type $\O(X,Y)$ are isomorphic and calculate their ranks.

In this paper, we consider the monoids of all endomorphisms, of all weak endomorphisms, of all strong endomorphisms and of all 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.

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.

Summary: We consider various classes of monoids of transformations on a finite chain, in particular of transformations that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested in computing both their cardinals and their idempotent numbers. Fibonacci and Lucas numbers play an essential role in the last computations.

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.