## Publications

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Book Chapter
Delgado, Manuel, and Vítor H. Fernandes. "Abelian kernels, solvable monoids and the abelian kernel length of a finite monoid." Semigroups and languages. World Sci. Publ., River Edge, NJ, 2004. 68-85.
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.
Conference Proceedings
Semigroups and formal languages. Eds. Jorge M. André, V{\'ı}tor H. Fernandes, Mário J. J. Branco, Gracinda M. S. Gomes, John Fountain, and John C. Meakin. Proceedings of the International Conference held at the Universidade de Lisboa, Lisboa, July 12–15, 2005. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.
Semigroups and languages. Eds. Isabel M. Araújo, Mário J. J. Branco, V{\'ı}tor H. Fernandes, and Gracinda M. S. Gomes. Proceedings of the workshop held at the University of Lisbon, Lisboa, November 27–29, 2002. River Edge, NJ: World Scientific Publishing Co. Inc., 2004.
Journal Article
Delgado, Manuel, and Vítor H. Fernandes. "Abelian kernels of monoids of order-preserving maps and of some of its extensions." Semigroup Forum. 68 (2004): 335-356.Website
Delgado, Manuel, and Vítor H. Fernandes. "Abelian kernels of some monoids of injective partial transformations and an application." Semigroup Forum. 61 (2000): 435-452.Website
Araújo, João, Vítor H. Fernandes, Manuel M. Jesus, Victor Maltcev, and James D. Mitchell. "Automorphisms of partial endomorphism semigroups." Publicationes Mathematicae Debrecen. 79.1-2 (2011): 23-39.
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., 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., 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. "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., 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, V. H., M. M. Jesus, V. Maltcev, and J. D. Mitchell. "Endomorphisms of the semigroup of order-preserving mappings." Semigroup Forum. 81 (2010): 277-285.Website
Fernandes, Vítor H. "The idempotent-separating degree of a block-group." Semigroup Forum. 76 (2008): 579-583.Website
André, J. M., V. H. Fernandes, and J. D. Mitchell. "Largest 2-generated subsemigroups of the symmetric inverse semigroup." Proc. Edinb. Math. Soc. (2). 50 (2007): 551-561.Website
Dimitrova, I., Vítor H. Fernandes, and J. Koppitz. "The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain." Publicationes Mathematicae Debrecen. 81.1-2 (2012): 11-29.
Fernandes, V. H. "The monoid of all injective order preserving partial transformations on a finite chain." Semigroup Forum. 62 (2001): 178-204.
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, Vitor H. "Normally ordered inverse semigroups." Semigroup Forum. 56 (1998): 418-433.Website
Fernandes, Vítor H. "Normally ordered semigroups." Glasg. Math. J.. 50 (2008): 325-333.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 $\mathcal{C}_2$.

Dimitrova, I., Vítor H. Fernandes, and J. Koppitz. "A note on generators of the endomorphism semigroup of an infinite countable chain." Journal of Algebra and its Applications (DOI: 10.1142/S0219498817500311). 16 (2017): 1750031 (9 pages). AbstractWebsite

In this note, we consider the semigroup $O(X)$ of all order endomorphisms of an infinite chain $X$ and the subset $J$ of $O(X)$ of all transformations $\alpha$ such that $|Im(\alpha)|=|X|$. For an infinite countable chain $X$, we give a necessary and sufficient condition on $X$ for $O(X) = < J >$ to hold. We also present a sufficient condition on $X$ for $O(X) = < J >$ to hold, for an arbitrary infinite chain $X$.

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