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V
Fernandes, Vítor H. "The Vagner-Preston representation of a block-group." Southeast Asian Bull. Math.. 45.6 (2021): 805-812. AbstractWebsite

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.

S
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.

Fernandes, Vitor H. "Semigroups of order preserving mappings on a finite chain: a new class of divisors." Semigroup Forum. 54 (1997): 230-236.Website
Semigroups and languages. Eds. Isabel M. Araújo, Mário J. J. Branco, V{\'ı}tor H. Fernandes, and Gracinda M. S. Gomes. Proceedings of the workshop held at the University of Lisbon, Lisboa, November 27–29, 2002. River Edge, NJ: World Scientific Publishing Co. Inc., 2004.
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.
R
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
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.

Dimitrova, I., Vítor H. Fernandes, J. Koppitz, and T. M. Quinteiro. "Ranks of monoids of endomorphisms of a finite undirected path (DOI: 10.1007/s40840-019-00762-4)." Bulletin of the Malaysian Mathematical Sciences Society. 43 (2020): 1623-1645. AbstractWebsite

In this paper we study the widely considered endomorphisms and weak endomorphisms of a finite undirected path from monoid generators perspective. Our main aim is to determine the ranks of the monoids $wEnd P_n$ and $End P_n$ of all weak endomorphisms and all endomorphisms of the undirected path $P_n$ with $n$ vertices. We also consider strong and strong weak endomorphisms of $P_n$.

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.

Caneco, Rita, Vítor H. Fernandes, and Teresa M. Quinteiro. "Ranks and presentations of some normally ordered inverse semigroups." Periodica Mathematica Hungarica (DOI 10.1007/s10998-022-00448-8). 85 (2022): 435-447. AbstractWebsite

In this paper we compute the rank and exhibit a presentation for the monoids
of all $P$-stable and $P$-order preserving partial permutations on a finite set
$\Omega$, with $P$ an ordered uniform partition of $\Omega$. These (inverse)
semigroups constitute a natural class of generators of the pseudovariety of
inverse semigroups ${\sf NO}$ of all normally ordered (finite) inverse
semigroups.

Fernandes, Vítor H., J. Koppitz, and T. Musunthia. "The rank of the semigroup of all order-preserving transformations on a finite fence." Bulletin of the Malaysian Mathematical Sciences Society (DOI: 10.1007/s40840-017-0598-1). 42.5 (2019): 2191-2211. AbstractWebsite

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.

P
Dimitrova, I., Vítor H. Fernandes, J. Koppitz, and T. M. Quinteiro. "Presentations for three remarkable submonoids of the dihedral inverse monoid on a finite set." Semigroup Forum (DOI 10.1007/s00233-023-10396-5; Online 31 Oct 2023). 107 (2023): 315-338. AbstractWebsite

In this paper we consider the submonoids OPDI_n, MDI_n and ODI_n of the dihedral inverse monoid DI_n of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids.

Fernandes, Vítor H. "Presentations for some monoids of partial transformations on a finite chain: a survey." Semigroups, algorithms, automata and languages (Coimbra, 2001). World Sci. Publ., River Edge, NJ, 2002. 363-378.
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
Fernandes, Vítor H., Gracinda M. S. Gomes, and Manuel M. Jesus. "Presentations for some monoids of injective partial transformations on a finite chain." Southeast Asian Bull. Math.. 28 (2004): 903-918.
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.

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.

Dimitrova, I., Vítor H. Fernandes, J. Koppitz, and T. M. Quinteiro. "Partial Automorphisms and Injective Partial Endomorphisms of a Finite Undirected Path." Semigroup Forum. 103 (2021): 87-105. AbstractWebsite

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 IEnd(P_n) and 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 PAut(P_n) and IEnd(P_n) and calculate their cardinals.

O
Fernandes, Vítor H. "Oriented transformations on a finite chain: another description." Commun. Korean Math. Soc. (DOI 10.4134/CKMS.c220272; Online 12 July 2023). 38.3 (2023): 725-731. AbstractWebsite

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.

Dimitrova, I., Vítor H. Fernandes, J. Koppitz, and T. M. Quinteiro. "On three submonoids of the dihedral inverse monoid on a finite set." Bulletin of the Malaysian Mathematical Sciences Society (DOI 10.1007/s40840-023-01620-0; Online 11 Dec 2023). 47 (2024): 27. AbstractWebsite

In this paper we consider three submonoids of the dihedral inverse monoid DI_n, namely its submonoids OPDI_n, MDI_n and ODI_n of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green's relations and determine the rank.

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.
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 Tânia Paulista. "On the Rank of Monoids of Endomorphisms of a Finite Directed Path." Asian-European Journal of Mathematics (DOI 10.1142/S1793557123500699; Online 28 Oct 2022). 16.04 (2023): 2350069 (13 pages). AbstractWebsite

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.

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. "On the monoid of partial isometries of a wheel graph." Asian-European Journal of Mathematics (DOI 10.1142/S1793557123502388; Online 16 Dec 2023). 17.1 (2024): 2350238 (18 pages). AbstractWebsite

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.