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Fernandes, Vitor H. "Normally ordered inverse semigroups." Semigroup Forum. 56 (1998): 418-433.Website
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 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 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.

Fernandes, Vítor H. "On monotone alternating inverse monoids." (Submitted). AbstractWebsite

In this paper, we consider the inverse submonoids AM_n of monotone transformations and AO_n of order-preserving transformations of the alternating inverse monoid AI_n on a chain with n elements. We compute the cardinalities, describe the Green's structures and the congruences, and calculate the ranks of these two submonoids of AI_n.

Fernandes, Vítor H. "The idempotent-separating degree of a block-group." Semigroup Forum. 76 (2008): 579-583.Website
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. "On the monoid of order-preserving transformations of a finite chain whose ranges are intervals." Semigroup Forum (DOI 10.1007/s00233-024-10466-2; Online 19 Aug 2024). 109.2 (2024): 336-346. AbstractWebsite

In this note we give a presentation for the monoid IO_n of all order-preserving transformations of a n-chain whose ranges are intervals. We also consider the submonoid IO_n^- of IO_n consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.

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 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 cyclic inverse monoid on a finite set." Asian-European Journal of Mathematics (DOI 10.1142/S1793557124500177; Online 6 Mar 2024). 17.2 (2024): 2450017 (16 pages). AbstractWebsite

In this paper we study the cyclic inverse monoid CI_n on a set Ω_n with n elements, i.e. the inverse submonoid of the symmetric inverse monoid on Ω_n consisting of all restrictions of the elements of a cyclic subgroup of order n acting cyclically on Ω_n. We show that CI_n has rank 2 (for n⩾2) and n⋅2^n−n+1 elements. Moreover, we give presentations of CI_n on n+1 generators and (n^2+3n+4)/2 relations and on 2 generators and (n^2−n+6)/2 relations. We also consider the remarkable inverse submonoid OCI_n of CI_n constituted by all its order-preserving transformations. We show that OCI_n has rank n and 3⋅2^n−2n−1 elements. Furthermore, we exhibit presentations of OCI_n on n+2 generators and (n^2+3n+8)/2 relations and on n generators and (n^2+3n)/2 relations.

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 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, V. H. "The monoid of all injective order preserving partial transformations on a finite chain." Semigroup Forum. 62 (2001): 178-204.
Fernandes, Vítor 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. "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., 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 oriented alternating inverse monoids." (Submitted). AbstractWebsite

In this paper, we consider the inverse submonoids AOR_n of oriented transformations and AOP_n of orientation-preserving transformations of the alternating inverse monoid AI_n on a chain with n elements. We compute the cardinalities, describe the Green's structures and the congruences, and calculate the ranks of AOR_n and AOP_n.

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., Preeyanuch Honyam, Teresa M. Quinteiro, and Boorapa Singha. "On semigroups of endomorphisms of a chain with restricted range." Semigroup Forum (DOI: 10.1007/s00233-013-9548-x). 89.1 (2014): 77-104. AbstractWebsite

Let $X$ be a finite or infinite chain and let $\O(X)$ be the monoid of all endomorphisms of $X$.
In this paper, we describe the largest regular subsemigroup of $\O(X)$ and Green's relations on $\O(X)$.
In fact, more generally, if $Y$ is a nonempty subset of $X$ and $\O(X,Y)$ is the subsemigroup of $\O(X)$ of all elements with range contained in $Y$,
we characterize the largest regular subsemigroup of $\O(X,Y)$ and Green's relations on $\O(X,Y)$.
Moreover, for finite chains, we determine when two semigroups of the type $\O(X,Y)$ are isomorphic and calculate their ranks.

Fernandes, Vítor H., J. Koppitz, and T. Musunthia. "Presentations for monoids of endomorphisms of a star graph." (Submitted). AbstractWebsite

In this paper, we consider the monoids of all endomorphisms, of all weak endomorphisms, of all strong endomorphisms and of all strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

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 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.