Presentations for monoids of finite partial isometries

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.Comment: 11 pages, submitte

Normally ordered semigroups

Glasgow Mathematical Journal, nº 50 (2008), p. 325-333In 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 blockgroups forms a pseudovariety of semigroups and, by using theMunn representation of a block-group, we deduce the decompositions in Mal’cev products NOS = EI m POI and NOS \ A = N 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 all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing that BG = EI m Ecom = N m Ecom, where BG and Ecom denote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively

The idempotent-separating degree of a block-group

Semigroup Forum, nº76 (2008), pg.579-583In this paper we describe the least non-negative 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

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

Funding Information: This work is funded by national funds through the FCT \u2013 Funda\u00E7\u00E3o para a Ci\u00EAncia e a Tecnologia, I.P., under the scope of the (Center for Mathematics and Applications) projects UIDB/00297/2020 ( https://doi.org/10.54499/UIDB/00297/2020 ) and UIDP/00297/2020 ( https://doi.org/10.54499/UIDP/00297/2020 ). Publisher Copyright: © The Author(s) 2024.We give a presentation for the monoid IOn of all order-preserving transformations of an n-chain whose ranges are intervals. We also consider the submonoid IOn- of IOn consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.publishersversionpublishe

Solvable monoids with commuting idempotents

International Journal of Algebra and Computation, 15, nº 3 (2005), p. 547-570The notion of Abelian kernel of a nite monoid extends the notion of derived subgroup of a nite group. In this line, an extension of the notion of solvable group to monoids is quite natural: they are the monoids such that the chain of Abelian kernels ends with the submonoid generated by the idempotents. We prove in this paper that the nite idempotent commuting monoids satisfying this property are precisely those whose subgroups are solvable

On semigroups of endomorphisms of a chain with restricted range

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.Comment: To appear in Semigroup Foru