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Publisher Copyright: © 2022 Taylor & Francis Group, LLC. In this paper we consider the monoid (Formula presented.) 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 (Formula presented.). We also describe Green’s relations of (Formula presented.) and calculate its cardinal. authorsversion published

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 [Formula: see text] of all weak endomorphisms of a directed path with [Formula: see text] vertices, which is a submonoid of the widely studied monoid [Formula: see text] of all order-preserving transformations of an [Formula: see text]-chain. Also, we describe the regular elements of [Formula: see text] and calculate its size and number of idempotents.

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.

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 intr...

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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 $$\mathrm {IEnd}(P_n)$$ and $$\mathrm {PAut}(P_n)$$ 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 $$\mathrm {PAut}(P_n)$$ and $$\mathrm {IEnd}(P_n)$$ and calculate their cardinals.

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

We characterize the monoids of endomorphisms of the semigroup of all order-preserving partial transformations and of the semigroup of all order-preserving partial permutations of a finite chain.

In this note we consider the monoid $\mathcal{PODI}_n$ of all monotone partial permutations on $\{1,\ldots,n\}$ and its submonoids $\mathcal{DP}_n$, $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $\mathcal{PODI}_n$ is a quotient of a semidirect product of $\mathcal{POI}_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $\mathcal{DP}_n$ is a quotient of a semidirect product of $\mathcal{ODP}_n$ and $\mathcal{C}_2$., Comment: 10 pages, submitted

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 [24], we complete the study of the ranks (and of idempotent ranks, when applicable) of all their ideals.

The objective of this paper is to study the monoid of all partial transformations of a finite set that preserve a uniform partition. In addition to proving that this monoid is a quotient of a wreath product with respect to a congruence relation, we show that it is generated by 5 generators, we compute its order and determine a presentation on a minimal generating set.

Let [Formula: see text] be the semigroup of all partial transformations on X, [Formula: see text] and [Formula: see text] be the subsemigroups of [Formula: see text] of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let [Formula: see text], [Formula: see text] and [Formula: see text]. In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green's relations of [Formula: see text]. In this paper, we present analogous results for both [Formula: see text] and [Formula: see text]. For a finite set X with |X| ≥ 3, the ranks of [Formula: see text], [Formula: see text] and [Formula: see text] are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of [Formula: see text], [Formula: see text] and [Formula: see text] for any proper non-empty subset Y of X.

We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of N, the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.

In this article we consider the monoid 𝒪 m × n of all order-preserving full transformations on a chain with mn elements that preserve a uniform m-partition and its submonoids and of all extensive transformations and of all co-extensive transformations, respectively. We determine their ranks and construct a bilateral semidirect product decomposition of 𝒪 m × n in terms of and .

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.

We show that if a semigroup T divides a semigroup of full order preserving transformations of a finite chain, then so does any semidirect product S⋊T where S is a finite semilattice whose natural order makes S a chain.

We characterize the endomorphisms of the semigroup of all order-preserving mappings on a finite chain. We show that there are three types of endomorphism: automorphisms, constants, and a certain type of endomorphism with two idempotents in the image.

In this note we characterize the least positive integer n such that there exists an idempotent-separating homomorphism from a finite block-group S into the monoid of all partial transformations of a set with n elements. In particular, as for a fundamental semigroup S this number coincides with the smallest size of a set for which S can be faithfully represented by partial transformations, we obtain a generalization of Easdown’s result established for fundamental finite inverse semigroups.

Let X n be a chain with n elements (n ∈ ℕ), and let 𝒪𝒫 n be the monoid of all orientation-preserving transformations of X n . In this article, for any nonempty subset Y of X n , we consider the subsemigroup 𝒪𝒫 n (Y) of 𝒪𝒫 n of all transformations with range contained in Y: We describe the largest regular subsemigroup of 𝒪𝒫 n (Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type 𝒪𝒫 n (Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid 𝒪ℛ n of all either orientation-preserving or orientation-reversing transformations of X n .

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

In this note, we consider the semigroup [Formula: see text] of all order endomorphisms of an infinite chain [Formula: see text] and the subset [Formula: see text] of [Formula: see text] of all transformations [Formula: see text] such that [Formula: see text]. For an infinite countable chain [Formula: see text], we give a necessary and sufficient condition on [Formula: see text] for [Formula: see text] to hold. We also present a sufficient condition on [Formula: see text] for [Formula: see text] to hold, for an arbitrary infinite chain [Formula: see text].

Glasgow Mathematical Journal, nº 47 (2005), pg. 413-424 This paper is mainly dedicated to describing the congruences on certain monoids of transformations on a finite chain Xn with n elements. Namely, we consider the monoids ODn and PODn of all full, respectively partial, transformations on Xn that preserve or reverse the order, as well as the submonoid POn of PODn of all its order-preserving elements. The inverse monoid PODIn of all injective elements of PODn is also considered. We show that in POn any congruence is a Rees congruence, but this may not happen in the monoids ODn, PODIn and PODn. However in all these cases the congruences form a chain. This work was developed within the activities of Centro de ´Algebra da Universidade de Lisboa, supported by FCT and FEDER, within project POCTI ”Fundamental and Applied Algebra”

In this paper we calculate presentations for some natural monoids of transformations on a chain X n = {1

Semigroup Forum vol. 68 (2004), p. 335–356 We compute the abelian kernels of some monoids of partial transformations. More precisely, the cases of the symmetric inverse monoid, monoids of order-preserving or order-reversing transformations and monoids of orientationpreserving or orientation-reversing transformations are treated.

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 $O(X,Y)$ 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., To appear in Semigroup Forum

In this paper we present a division theorem for the pseudovariety of semigroups OP generated by all semigroups of orientation preserving full transformations on a finite chain, which is achieved by using a natural representation of a certain monoid of transformations of a set X as a monoid of transformations of a subset of X.

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S α (α∈ Y ) is finitely generated/presented if and only if Y is finite and all S α are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups.

In this paper we compute the abelian kernels of the monoids POI n and POPI n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI n (n epsilon N) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI n (n epsilon N) with the pseudovariety Ab of all finite abelian groups.

The rank of a semigroup, an important and relevant concept in Semigroup Theory, is the cardinality of a least-size generating set. Semigroups of transformations that preserve or reverse the order or the orientation as well as semigroups of transformations preserving an equivalence relation have been widely studied over the past decades by many authors. The purpose of this article is to compute the ranks of the monoid 𝒪ℛ m×n of all orientation-preserving or orientation-reversing full transformations on a chain with mn elements that preserve a uniform m-partition and of its submonoids 𝒪𝒫 m×n of all orientation-preserving transformations and 𝒪𝒟 m×n of all order-preserving or order-reversing full transformations. These three monoids are natural extensions of 𝒪 m×n , the monoid of all order-preserving full transformations on a chain with mn elements that preserve a uniform m-partition. Given m, n ≥ 2, we show that the rank of 𝒪𝒫 m×n is equal to , for m > 2, and equal to , for m = 2, the rank of 𝒪𝒟 m×n is equal...

In this paper we aim to prove that every semigroup of the pseudovariety generated by all semigroups of partial, injective and order preserving transformations on a finite chain belongs to the pseudovariety generated by all semigroups of order preserving mappings on a finite chain.

The study of the semigroups OPn, of all orientation-preserving transformations on an n-element chain, and ORn, of all orientation-preserving or orientation-reversing transformations on an n-element chain, has began in [17] and [5]. In order to bring more insight into the subsemigroup structure of OPn and ORn, we characterize their maximal subsemigroups.

The main subject of this paper is the description of the congruences on certain monoids of transformations on a finite chain X n with n elements. Namely, we consider the monoids OR n and POR n of all full, respectively partial, transformations on X n that preserve or reverse the orientation, as well as their respective submonoids OP n and POP n of all orientation-preserving elements. The inverse monoid PORI n of all injective elements of POR n is also considered.

In this paper we introduce the notion of normally ordered block-group as a natural extension of the notion of normally ordered inverse semigroup considered previously by the author. We prove that the class NOS of all normally ordered block-groups forms a pseudovariety of semigroups and, by using the Munn representation of a block-group, we deduce the decompositions in Mal'cev products NOS = EI$\petite m$POI and NOS ∩ A = N$\petite m$POI, where A, EI and N denote the pseudovarieties of all aperiodic semigroups, all semigroups with just one idempotent and all nilpotent semigroups, respectively, and POI denotes the pseudovariety of semigroups generated by all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing the equalities BG = EI$\petite m$Ecom = N$\petite m$Ecom, where BG and Ecom denote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively.

The symmetric inverse monoid $\mathcal{I}_{n}$ is the set of all partial permutations of an $n$-element set. The largest possible size of a $2$-generated subsemigroup of $\mathcal{I}_{n}$ is determined. Examples of semigroups with these sizes are given. Consequently, if $M(n)$ denotes this maximum, it is shown that $M(n)/|\mathcal{I}_{n}|\rightarrow1$ as $n\rightarrow\infty$. Furthermore, we deduce the known fact that $\mathcal{I}_{n}$ embeds as a local submonoid of an inverse $2$-generated subsemigroup of $\mathcal{I}_{n+1}$.

We consider the symmetric inverse monoid ℐn of an n-element chain and its inverse submonoids ℐn, ℐn, ℐn and ℘ℐn of all order-preserving, order-preserving or order-reversing, orientation-preserving and orientation-preserving or orientation-reversing transformations, respectively, and give descriptions of their Abelian kernels relative to decidable pseudovarieties of Abelian groups.

In this paper we do not make a clear distinction between what is introduction and preliminaries. In fact, we have decided to put it all in a single section which is divided into 3 subsections. The first, concerning kernels of finite monoids and related properties, should mainly serve as a general motivation for the study that we will do later with some particular classes of monoids. In the second subsection are introduced those classes of monoids.

In this paper we study several structural properties of the monoids \poi n of all injective order preserving partial transformations on a chain with n elements. Our main aim is to give a presentation for these monoids.

We show that if S is a finite semigroup with aperiodic idempotent-generated subsemigroup and H is a pseudovariety of groups, then the sequence of iterated H-kernels of S stops at the idempotent-generated subsemigroup if and only if each subgroup of S belongs to the wreath product closure of H. Applications are given to Mal'cev products.