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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.

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Li, De Biao, and Vítor H. Fernandes. "Endomorphisms of semigroups of oriented transformations." Semigroup Forum (DOI 10.1007/s00233-022-10325-y; Online 2 Dec 2022). 106 (2023): 184-210. AbstractWebsite

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.

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Fernandesh, V. U. "A new class of divisors of semigroups of isotone mappings of finite chains." Izv. Vyssh. Uchebn. Zaved. Mat. (2002): 51-59.
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., 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., 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.
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.

Fernandes, Vítor H. "The idempotent-separating degree of a block-group." Semigroup Forum. 76 (2008): 579-583.Website
Fernandes, Vitor H. "Normally ordered inverse semigroups." Semigroup Forum. 56 (1998): 418-433.Website
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.

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
Fernandes, Vítor H. "Oriented transformations on a finite chain: another description." Commun. Korean Math. Soc. (In Press). Abstract

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.

Fernandes, Vítor H., and Tânia Paulista. "On the monoid of partial isometries of a finite star graph." Communications in Algebra (DOI 10.1080/00927872.2022.2121404; Online 14 Sep 2022). 51.3 (2023): 1028-1048. AbstractWebsite

In this paper we consider the monoid DPSn 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 DPSn. We also describe Green’s relations of DPSn and calculate its cardinal.

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., 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., 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 Paulo G. Santos. "Endomorphisms of semigroups of order-preserving partial transformations." Semigroup Forum (10.1007/s00233-018-9948-z). 99 (2019): 333-344. AbstractWebsite

In this paper we characterize the monoids of endomorphisms of the semigroups PO_n and POI_n of all order-preserving partial transformations and of all order-preserving partial permutations, respectively, of a finite n-chain.

Fernandes, Vitor H. "Semigroups of order preserving mappings on a finite chain: a new class of divisors." Semigroup Forum. 54 (1997): 230-236.Website
Fernandes, Vítor H., and Teresa M. Quinteiro. "A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutations." Bull. Korean Math. Soc.. 53.2 (2016): 495-506. AbstractWebsite

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 $\mathcal{C}_2$.

Fernandes, Vítor H. "On divisors of pseudovarieties generated by some classes of full transformation semigroups." Algebra Colloq.. 15 (2008): 581-588.
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
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. "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.

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.