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.
AbstractA 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., and Teresa M. Quinteiro. "
Bilateral semidirect product decompositions of transformation monoids."
Semigroup Forum. 82 (2011): 271-287.
AbstractSummary: 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 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.
AbstractIn 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.