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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. AbstractWebsite

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.

F
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., 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., 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., 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. "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., 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., and Jintana Sanwong. "On the rank of semigroups of transformations on a finite set with restricted range." Algebra Colloquium. 21.3 (2014): 497-510.
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, 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 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, 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. "On divisors of pseudovarieties generated by some classes of full transformation semigroups." Algebra Colloq.. 15 (2008): 581-588.
Fernandes, V. 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. "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, V. H., M. M. Jesus, and B. Singha. "On orientation-preserving transformations of a chain." Communications in Algebra (DOI 10.1080/00927872.2020.1870996). 49.6 (2021): 2300-2325. AbstractWebsite

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 introduced in 1998 by McAlister and, independently, in 1999 by Catarino and Higgins.
We consider the monoid POP(X) of all orientation-preserving partial transformations on a finite or infinite chain X and its submonoids OP(X) and POPI(X) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on X, respectively.
The monoid PO(X) of all order-preserving partial transformations on X and its injective counterpart POI(X) are also considered.
We study the regularity and give descriptions of the Green's relations of the monoids POP(X), PO(X), OP(X), POPI(X) and POI(X).

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.
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., Preeyanuch Honyam, Teresa M. Quinteiro, and Boorapa Singha. "On semigroups of orientation-preserving transformations with restricted range." Communications in Algebra (DOI:10.1080/00927872.2014.975345). 44.1 (2016): 253-264. AbstractWebsite

Let $X_n$ be a chain with n elements ($n\in\N$) and let $\OP_n$ be the monoid of all orientation-preserving transformations of $X_n$. In this paper, 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, and Green's relations on $\OP_n(Y)$. 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 $X_n$.

Fernandes, Vítor H. "Normally ordered semigroups." Glasg. Math. J.. 50 (2008): 325-333.Website