Publications

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2016
Fernandes, Vítor H., and Teresa M. Quinteiro. "Presentations for monoids of finite partial isometries." Semigroup Forum (DOI: 10.1007/s00233-015-9759-4). 93.1 (2016): 97-110. AbstractWebsite

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.

2015
Cicalò, Serena, Vítor H. Fernandes, and Csaba Schneider. "Partial transformation monoids preserving a uniform partition." Semigroup Forum (DOI 10.1007/s00233-014-9629-5). 90.2 (2015): 532-544. AbstractWebsite

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.

Zhao, Ping, and Vítor H. Fernandes. "The ranks of ideals in various transformation monoids." Communications in Algebra (DOI:10.1080/00927872.2013.847946) . 43.2 (2015): 674-692. Abstractauthorsfinalversion.pdfWebsite

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.

2014
Fernandes, Vítor H., Preeyanuch Honyam, Teresa M. Quinteiro, and Boorapa Singha. "On semigroups of endomorphisms of a chain with restricted range." Semigroup Forum (DOI: 10.1007/s00233-013-9548-x). 89.1 (2014): 77-104. AbstractWebsite

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.

Fernandes, Vítor H., and Jintana Sanwong. "On the rank of semigroups of transformations on a finite set with restricted range." Algebra Colloquium. 21.3 (2014): 497-510.authorsfinalversion.pdfWebsite
Fernandes, Vítor H., and Teresa M. Quinteiro. "On the ranks of certain monoids of transformations that preserve a uniform partition." Communications in Algebra. 42.2 (2014): 615-636.
2013
Delgado, Manuel, and Vítor H. Fernandes. "Rees quotients of numerical semigroups." Portugaliae Mathematica. 70.2 (2013): 93-112. AbstractWebsite

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.

2012
Fernandes, Vítor H., and Teresa M. Quinteiro. "The cardinal of various monoids of transformations that preserve a uniform partition." Bulletin of the Malaysian Mathematical Sciences Society. 35.4 (2012): 885-896.
Dimitrova, I., Vítor H. Fernandes, and J. Koppitz. "The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain." Publicationes Mathematicae Debrecen. 81.1-2 (2012): 11-29.
2011
Araújo, João, Vítor H. Fernandes, Manuel M. Jesus, Victor Maltcev, and James D. Mitchell. "Automorphisms of partial endomorphism semigroups." Publicationes Mathematicae Debrecen. 79.1-2 (2011): 23-39.
Fernandes, Vítor H., Gracinda M. S. Gomes, and Manuel M. Jesus. "The cardinal and the idempotent number of various monoids of transformations on a finite chain." Bulletin of the Malaysian Mathematical Sciences Society. 34.2 (2011): 79-85. Abstract

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.

Fernandes, Vítor H., and Teresa M. Quinteiro. "On the monoids of transformations that preserve the order and a uniform partition." Communications in Algebra. 39.8 (2011): 2798-2815.
Fernandes, Vítor H., and Teresa M. Quinteiro. "Bilateral semidirect product decompositions of transformation monoids." Semigroup Forum. 82 (2011): 271-287. Abstract
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.
2010
Fernandes, Vítor H., M. M. Jesus, V. Maltcev, and J. D. Mitchell. "Endomorphisms of the semigroup of order-preserving mappings." Semigroup Forum. 81 (2010): 277-285.Website
Fernandes, Vítor H., and M. V. Volkov. "On divisors of semigroups of order-preserving mappings of a finite chain." Semigroup Forum. 81 (2010): 551-554.Website
2009
Fernandes, Vítor H., Gracinda M. S. Gomes, and Manuel M. Jesus. "Congruences on monoids of transformations preserving the orientation of a finite chain." J. Algebra. 321 (2009): 743-757.Website
2008
Fernandes, Vítor H. "The idempotent-separating degree of a block-group." Semigroup Forum. 76 (2008): 579-583.Website
Fernandes, Vítor H. "Normally ordered semigroups." Glasg. Math. J.. 50 (2008): 325-333.Website
Fernandes, Vítor H. "On divisors of pseudovarieties generated by some classes of full transformation semigroups." Algebra Colloq.. 15 (2008): 581-588.
2007
Semigroups and formal languages. Eds. Jorge M. André, V{\'ı}tor H. Fernandes, Mário J. J. Branco, Gracinda M. S. Gomes, John Fountain, and John C. Meakin. Proceedings of the International Conference held at the Universidade de Lisboa, Lisboa, July 12–15, 2005. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.
André, J. M., V. H. Fernandes, and J. D. Mitchell. "Largest 2-generated subsemigroups of the symmetric inverse semigroup." Proc. Edinb. Math. Soc. (2). 50 (2007): 551-561.Website
2006
Cordeiro, E., M. Delgado, and V. H. Fernandes. "Relative abelian kernels of some classes of transformation monoids." Bull. Austral. Math. Soc.. 73 (2006): 375-404.Website
2005
Fernandes, Vítor H., Gracinda M. S. Gomes, and Manuel M. Jesus. "Congruences on monoids of order-preserving or order-reversing transformations on a finite chain." Glasg. Math. J.. 47 (2005): 413-424.Website
Fernandes, Vítor H., Gracinda M. S. Gomes, and Manuel M. Jesus. "Presentations for some monoids of partial transformations on a finite chain." Comm. Algebra. 33 (2005): 587-604.Website
Delgado, Manuel, and Vítor H. Fernandes. "Solvable monoids with commuting idempotents." Int. J. Algebra Comput.. 15 (2005): 547-570. Abstract

The notion of the Abelian kernel of a finite monoid is a generalization of that of the derived subgroup of a finite group. A monoid $M$ is then called solvable if its chain of Abelian kernels terminates with the submonoid of $M$ generated by its idempotents. The main result of this paper is that the finite idempotent commuting monoids bearing this property are precisely those whose subgroups are solvable. In particular any finite aperiodic monoid is Abelian-solvable in this sense. A generalization of the main result of this paper has been published [in Int. J. Algebra Comput. 14, No. 5-6, 655-665 (2004; Zbl 1081.20067)] by the authors and ıt S. Margolis and ıt B. Steinberg.