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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.

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

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.

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 $P_n$.

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.

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