@article {16806, title = {Groups of permutations preserving orientation (parity) of subsets of a fixed size, and related monoids}, year = {Submitted}, abstract = {

We study permutations on n elements preserving orientation (parity) of every subset of size k. We describe all groups of these permutations. Unexpectedly, these groups (except for some special cases) are either trivial, cyclic or dihedral. In this context, we define and study monoids generalizing monoids of order-preserving mappings and monoids of orientation-preserving mappings.

}, url = {http://arxiv.org/abs/2402.05777}, author = {Fernandes, V{\'\i}tor H. and Vernitski, A.} } @article {16869, title = {On monoids of monotone partial transformations of a finite chain whose domains and ranges are intervals}, year = {Submitted}, abstract = {

In this note, we consider the monoid PIM_n of all partial monotone transformations on a chain with n elements whose domains and ranges are intervals and its submonoid IM_n constituted by the full transformations. For both of these monoids, our aim is to determine their cardinalities and ranks and define them by means of presentations. We also calculate the number of nilpotent elements of PIM_n.

}, url = {https://arxiv.org/abs/2504.19618}, author = {Ay{\i}k, H. and Fernandes, V{\'\i}tor H. and Korkmaz, E.} } @article {16828, title = {On monotone alternating inverse monoids}, year = {Submitted}, abstract = {

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{\textquoteright}s structures and the congruences, and calculate the ranks of these two submonoids of AI_n.

}, url = {https://arxiv.org/abs/2503.00820}, author = {Fernandes, V{\'\i}tor H.} } @article {16829, title = {On oriented alternating inverse monoids}, year = {Submitted}, abstract = {

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{\textquoteright}s structures and the congruences, and calculate the ranks of AOR_n and AOP_n.

}, url = {https://arxiv.org/abs/2503.00832}, author = {Fernandes, V{\'\i}tor H.} } @article {16626, title = {On semigroups of orientation-preserving partial permutations with restricted range}, year = {Submitted}, abstract = {

Let $\Omega_n$ be a finite chain with $n$ elements $(n\in\mathbb{N})$, and let $\mathcal{POPI}_{n}$ be the semigroup of all injective orientation-preserving partial transformations of $\Omega_n$. In this paper, for any nonempty subset $Y$ of $\Omega_n$, we consider the subsemigroup $\mathcal{POPI}_{n}(Y)$ of $\mathcal{POPI}_{n}$ of all transformations with range contained in $Y$. We describe the Green{\textquoteright}s relations and study the regularity of $\mathcal{POPI}_{n}(Y)$. Moreover, we calculate the rank of $\mathcal{POPI}_{n}(Y)$ and determine when two semigroups of this type are isomorphic.

}, url = {https://doi.org/10.48550/arXiv.2401.02185}, author = {De Biao Li and Fernandes, V{\'\i}tor H.} } @article {16855, title = {On the monoid of partial order-preserving transformations of a finite chain whose domains and ranges are intervals}, year = {Submitted}, abstract = {

In this paper, we consider the monoid PIO_n, of all partial order-preserving transformations on a chain with n elements whose domains and ranges are intervals, along with its submonoid PIO_n^- of order-decreasing transformations. Our main aim is to give presentations for PIO_n^- and PIO_n. Moreover, for both monoids, we describe regular elements and determine their ranks, cardinalities and the numbers of idempotents and nilpotents.

}, url = {https://arxiv.org/abs/2503.19459}, author = {Ay{\i}k, H. and Fernandes, V{\'\i}tor H. and Korkmaz, E.} } @article {16792, title = {Presentations for monoids of partial endomorphisms of a star graph}, year = {Submitted}, abstract = {

In this paper, we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial 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.

}, url = {https://arxiv.org/abs/2410.21084}, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J.} } @article {16797, title = {Presentations for monoids of endomorphisms of a star graph}, journal = {Asian-European Journal of Mathematics (DOI 10.1142/S1793557125500494)}, volume = {18}, year = {2025}, pages = {2550049}, abstract = {

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.

}, url = {https://doi.org/10.1142/S1793557125500494}, author = {Fernandes, V{\'\i}tor H. and Koppitz, J. and Musunthia, T.} } @article {16659, title = {Corrigendum on "Oriented transformations on a finite chain: another description" [Commun. Korean Math. Soc. 38 (2023), No. 3, pp. 725-731]}, journal = {Commun. Korean Math. Soc. (DOI 10.4134/CKMS.c240008; Online 12 July 2024) }, volume = {39}, year = {2024}, pages = {643-645}, abstract = {

In this note, we aim to correct some of the results presented in [1]. Namely, the statements of Proposition 2.1, Corollary 2.2, Corollary 2.3, Theorem 2.4 and Theorem 2.6, concerning only the monoids OP_n and POP_n, have to exclude transformations of rank two. All other results of [1], as well as those mentioned above but for the monoids OR_n and POR_n, do not require correction.

[1] V.H. Fernandes, Oriented transformations on a finite chain: another description, Commun. Korean Math. Soc. 38 (2023), 725-731.

}, url = {https://doi.org/10.4134/CKMS.c240008}, author = {Fernandes, V{\'\i}tor H.} } @article {16439, title = {Endomorphisms of semigroups of monotone transformations}, journal = {Journal of Algebra and its Applications (DOI 10.1142/S0219498824502244; Online 5 July 2023)}, volume = {23}, year = {2024}, pages = {2450224 (17 pages)}, abstract = {

In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

}, url = {https://doi.org/10.1142/S0219498824502244}, author = {De Biao Li and Fernandes, V{\'\i}tor H.} } @article {16594, title = {On monoids of endomorphisms of a cycle graph}, journal = {Mathematica Slovaca (DOI 10.1515/ms-2024-0078; Online 15 October 2024)}, volume = {74}, year = {2024}, pages = {1071-1088}, abstract = {

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green{\textquoteright}s relations and regularity of these monoids and calculate their cardinalities.

}, url = {https://www.degruyter.com/document/doi/10.1515/ms-2024-0078/html}, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J. and T.M. Quinteiro} } @article {16644, title = {On partial endomorphisms of a star graph}, journal = {Quaestiones Mathematicae (DOI 10.2989/16073606.2024.2374796; Online 31 July 2024)}, volume = {47}, year = {2024}, pages = {2485-2505}, abstract = {

In this paper we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to determine their ranks. We also describe their Green{\textquoteright}s relations, calculate their cardinalities and study their regularity.

}, url = {https://doi.org/10.2989/16073606.2024.2374796}, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J.} } @article {16548, title = {On the cyclic inverse monoid on a finite set}, journal = {Asian-European Journal of Mathematics (DOI 10.1142/S1793557124500177; Online 6 Mar 2024)}, volume = {17}, year = {2024}, pages = {2450017 (16 pages)}, abstract = {

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.

}, url = {https://doi.org/10.1142/S1793557124500177}, author = {Fernandes, V{\'\i}tor H.} } @article {16768, title = {On the monoid of order-preserving transformations of a finite chain whose ranges are intervals}, journal = {Semigroup Forum (DOI 10.1007/s00233-024-10466-2; Online 19 Aug 2024)}, volume = {109}, year = {2024}, pages = {336-346}, abstract = {

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.

}, url = {https://doi.org//10.1007/s00233-024-10466-2}, author = {Fernandes, V{\'\i}tor H.} } @article {16551, title = {On the monoid of partial isometries of a wheel graph}, journal = {Asian-European Journal of Mathematics (DOI 10.1142/S1793557123502388; Online 16 Dec 2023)}, volume = {17}, year = {2024}, pages = {2350238 (18 pages)}, abstract = {

In this paper, we consider the monoid DPW_n of all partial isometries of a wheel graph W_n with n+1 vertices. Our main objective is to determine the rank of DPW_n. In the process, we also compute the ranks of three notable subsemigroups of DPW_n. We also describe Green{\textquoteright}s relations of DPW_n and of its three considered subsemigroups.

}, url = {https://doi.org/10.1142/S1793557123502388}, author = {Fernandes, V{\'\i}tor H.} } @article {16549, title = {On three submonoids of the dihedral inverse monoid on a finite set}, journal = {Bulletin of the Malaysian Mathematical Sciences Society (DOI 10.1007/s40840-023-01620-0; Online 11 Dec 2023)}, volume = {47}, year = {2024}, pages = {27}, abstract = {

In this paper we consider three submonoids of the dihedral inverse monoid DI_n, namely its submonoids OPDI_n, MDI_n and ODI_n of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green{\textquoteright}s relations and determine the rank.

}, url = {https://doi.org/10.1007/s40840-023-01620-0}, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J. and T.M. Quinteiro} } @article {16191, title = {Endomorphisms of semigroups of oriented transformations}, journal = {Semigroup Forum (DOI 10.1007/s00233-022-10325-y; Online 2 Dec 2022)}, volume = {106}, year = {2023}, pages = {184{\textendash}210}, abstract = {

In this paper, we characterize the monoid of endomorphisms of the semigroup of all oriented full transformations of a finite chain, as well as the monoid of endomorphisms of the semigroup of all oriented partial transformations and the monoid of endomorphisms of the semigroup of all oriented partial permutations of a finite chain. Characterizations of the monoids of endomorphisms of the subsemigroups of all orientation-preserving transformations of the three semigroups aforementioned are also given. In addition, we compute the number of endomorphisms of each of these six semigroups.

}, url = {https://rdcu.be/c0TKs}, author = {De Biao Li and Fernandes, V{\'\i}tor H.} } @article {16535, title = {On the monoid of partial isometries of a cycle graph}, journal = {Turkish Journal of Mathematics (DOI 10.55730/1300-0098.3460)}, volume = {47}, year = {2023}, pages = {1746-1760}, abstract = {

In this paper we consider the monoid DPC_n of all partial isometries of a n-cycle graph C_n. We show that DPC_n is the submonoid of the monoid of all oriented partial permutations on a n-chain whose elements are precisely all restrictions of a dihedral group of order 2n. Our main aim is to exhibit a presentation of DPC_n. We also describe Green{\textquoteright}s relations of DPC_n and calculate its cardinality and rank.

}, url = {https://journals.tubitak.gov.tr/math/vol47/iss6/10/}, author = {Fernandes, V{\'\i}tor H. and Paulista, T{\^a}nia} } @article {15860, title = {On the monoid of partial isometries of a finite star graph}, journal = {Communications in Algebra (DOI 10.1080/00927872.2022.2121404; Online 14 Sep 2022)}, volume = {51}, year = {2023}, pages = {1028-1048}, abstract = {

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{\textquoteright}s relations of DPSn and calculate its cardinal.

}, url = {https://doi.org/10.1080/00927872.2022.2121404}, author = {Fernandes, V{\'\i}tor H. and Paulista, T{\^a}nia} } @article {15880, title = {On the Rank of Monoids of Endomorphisms of a Finite Directed Path}, journal = {Asian-European Journal of Mathematics (DOI 10.1142/S1793557123500699; Online 28 Oct 2022)}, volume = {16}, year = {2023}, pages = {2350069 (13 pages)}, abstract = {

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.

}, url = {https://doi.org/10.1142/S1793557123500699}, author = {Fernandes, V{\'\i}tor H. and Paulista, T{\^a}nia} } @article {16235, title = {Oriented transformations on a finite chain: another description}, journal = {Commun. Korean Math. Soc. (DOI 10.4134/CKMS.c220272; Online 12 July 2023)}, volume = {38}, year = {2023}, pages = {725-731}, abstract = {

Following the new description of an oriented full transformation on a finite chain given recently by Higgins and Vernitsk,
in this short note we present a refinement of this description which is extendable to partial transformations and to injective partial transformations.

}, url = {https://ckms.kms.or.kr/journal/view.html?doi=10.4134/CKMS.c220272}, author = {Fernandes, V{\'\i}tor H.} } @article {16550, title = {Presentations for three remarkable submonoids of the dihedral inverse monoid on a finite set}, journal = {Semigroup Forum (DOI 10.1007/s00233-023-10396-5; Online 31 Oct 2023)}, volume = {107}, year = {2023}, pages = {315-338}, abstract = {

In this paper we consider the submonoids OPDI_n, MDI_n and ODI_n of the dihedral inverse monoid DI_n of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids.

}, url = {https://doi.org/10.1007/s00233-023-10396-5}, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J. and T.M. Quinteiro} } @article {13242, title = {Ranks and presentations of some normally ordered inverse semigroups}, journal = {Periodica Mathematica Hungarica (DOI 10.1007/s10998-022-00448-8)}, volume = { 85}, year = {2022}, pages = {435{\textendash}447}, abstract = {

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
semigroups.

}, url = {https://doi.org/10.1007/s10998-022-00448-8}, author = {Caneco, Rita and Fernandes, V{\'\i}tor H. and Quinteiro, Teresa M.} } @article {12432, title = {On orientation-preserving transformations of a chain}, journal = {Communications in Algebra (DOI 10.1080/00927872.2020.1870996)}, volume = {49}, year = {2021}, pages = {2300-2325}, abstract = {

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{\textquoteright}s relations of the monoids POP(X), PO(X), OP(X), POPI(X) and POI(X).

}, url = {https://doi.org/10.1080/00927872.2020.1870996}, author = {Fernandes, V{\'\i}tor H. and Jesus, M. M. and B. Singha} } @article {15230, title = {Partial Automorphisms and Injective Partial Endomorphisms of a Finite Undirected Path}, journal = {Semigroup Forum}, volume = {103}, year = {2021}, pages = {87-105}, abstract = {

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids IEnd(P_n) and PAut(P_n) of all injective partial endomorphisms and of all partial automorphisms of the undirected path P_n with n vertices. We also describe Green{\textquoteright}s relations of PAut(P_n) and IEnd(P_n) and calculate their cardinals.

}, url = {https://link.springer.com/article/10.1007/s00233-021-10193-y}, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J. and T.M. Quinteiro} } @article {15210, title = {The Vagner-Preston representation of a block-group}, journal = {Southeast Asian Bull. Math.}, volume = {45}, year = {2021}, pages = {805-812}, abstract = {

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.

}, url = {http://www.seams-bull-math.ynu.edu.cn/}, author = {Fernandes, V{\'\i}tor H.} } @article {12789, title = {Ranks of monoids of endomorphisms of a finite undirected path (DOI: 10.1007/s40840-019-00762-4)}, journal = {Bulletin of the Malaysian Mathematical Sciences Society}, volume = {43}, year = {2020}, pages = {1623{\textendash}1645}, abstract = {

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 $P_n$.

}, url = {https://rdcu.be/bxTbr}, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J. and T.M. Quinteiro} } @article {12358, title = {Endomorphisms of semigroups of order-preserving partial transformations}, journal = {Semigroup Forum (10.1007/s00233-018-9948-z)}, volume = {99}, year = {2019}, pages = {333-344}, abstract = {

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.

}, url = {https://rdcu.be/YeoC}, author = {Fernandes, V{\'\i}tor H. and Paulo G Santos} } @article {11289, title = {The rank of the semigroup of all order-preserving transformations on a finite fence}, journal = {Bulletin of the Malaysian Mathematical Sciences Society (DOI: 10.1007/s40840-017-0598-1)}, volume = {42}, year = {2019}, pages = {2191-2211}, abstract = {

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.

}, url = {http://rdcu.be/FYNs}, author = {Fernandes, V{\'\i}tor H. and Koppitz, J. and Musunthia, T.} } @article {7513, title = {A note on generators of the endomorphism semigroup of an infinite countable chain}, journal = {Journal of Algebra and its Applications (DOI: 10.1142/S0219498817500311)}, volume = {16}, year = {2017}, pages = {1750031 (9 pages)}, abstract = {

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 $X$.

}, url = {http://dx.doi.org/10.1142/S0219498817500311 }, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J.} } @article {7088, title = {A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutations}, journal = {Bull. Korean Math. Soc.}, volume = {53}, year = {2016}, pages = {495-506}, abstract = {

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 $\mathcal{C}_2$.

}, url = {http://pdf.medrang.co.kr/kms01/BKMS/53/BKMS-53-2-495-506.pdf}, author = {Fernandes, V{\'\i}tor H. and Quinteiro, Teresa M.} } @article {5763, title = {On semigroups of orientation-preserving transformations with restricted range}, journal = {Communications in Algebra (DOI:10.1080/00927872.2014.975345)}, volume = {44}, year = {2016}, pages = {253-264}, abstract = {

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{\textquoteright}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 $X_n$.

}, url = {http://www.tandfonline.com/doi/pdf/10.1080/00927872.2014.975345}, attachments = {https://docentes.fct.unl.pt/sites/default/files/vhf/files/opnyv2.pdf}, author = {Fernandes, V{\'\i}tor H. and Preeyanuch Honyam and Quinteiro, Teresa M. and Boorapa Singha} } @article {6488, title = {Presentations for monoids of finite partial isometries}, journal = {Semigroup Forum (DOI: 10.1007/s00233-015-9759-4)}, volume = {93}, year = {2016}, pages = {97-110}, abstract = {

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.

}, url = {https://rdcu.be/5f5y}, author = {Fernandes, V{\'\i}tor H. and Quinteiro, Teresa M.} } @article {2813, title = {Partial transformation monoids preserving a uniform partition}, journal = {Semigroup Forum (DOI 10.1007/s00233-014-9629-5)}, volume = {90}, year = {2015}, pages = {532-544}, abstract = {

The objective of this paper is to study the monoid of all partial
transformations of a finite set that preserve a uniform partition. In addition
to proving that this monoid is a quotient of a wreath product with respect to a
congruence relation, we show that it is generated by 5 generators, we compute
its order and determine a presentation on a minimal generating set.

}, url = {http://link.springer.com/article/10.1007/s00233-014-9629-5}, author = {Cical{\`o}, Serena and Fernandes, V{\'\i}tor H. and Schneider, Csaba} } @article {2276, title = {The ranks of ideals in various transformation monoids}, journal = {Communications in Algebra (DOI:10.1080/00927872.2013.847946) }, volume = {43}, year = {2015}, pages = {674-692}, abstract = {

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.

}, url = {http://www.tandfonline.com/doi/abs/10.1080/00927872.2013.847946?journalCode=lagb20$\#$.VOhfX0L3-FQ}, attachments = {https://docentes.fct.unl.pt/sites/default/files/vhf/files/ranksv2.pdf}, author = {Ping Zhao and Fernandes, V{\'\i}tor H.} } @article {2620, title = {On semigroups of endomorphisms of a chain with restricted range}, journal = {Semigroup Forum (DOI: 10.1007/s00233-013-9548-x)}, volume = {89}, year = {2014}, pages = {77-104}, abstract = {

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{\textquoteright}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{\textquoteright}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.

}, url = {http://link.springer.com/article/10.1007\%2Fs00233-013-9548-x}, author = {Fernandes, V{\'\i}tor H. and Preeyanuch Honyam and Quinteiro, Teresa M. and Boorapa Singha} } @article {474, title = {On the rank of semigroups of transformations on a finite set with restricted range}, journal = {Algebra Colloquium}, volume = {21}, year = {2014}, note = {

DOI: 10.1142/S1005386714000431

}, pages = {497-510}, url = {https://doi.org/10.1142/S1005386714000431}, attachments = {https://docentes.fct.unl.pt/sites/default/files/vhf/files/rankrestrirange61.pdf}, author = {Fernandes, V{\'\i}tor H. and Sanwong, Jintana} } @article {476, title = {On the ranks of certain monoids of transformations that preserve a uniform partition}, journal = {Communications in Algebra}, volume = {42}, year = {2014}, pages = {615-636}, author = {Fernandes, V{\'\i}tor H. and Quinteiro, Teresa M.} } @article {2812, title = {Rees quotients of numerical semigroups}, journal = {Portugaliae Mathematica}, volume = {70}, year = {2013}, note = {

DOI: 10.4171/PM/1927

}, pages = {93-112}, abstract = {

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.

}, url = {http://www.ems-ph.org/journals/show_abstract.php?issn=0032-5155\&vol=70\&iss=2\&rank=1}, author = {Delgado, Manuel and Fernandes, V{\'\i}tor H.} } @article {472, title = {The cardinal of various monoids of transformations that preserve a uniform partition}, journal = {Bulletin of the Malaysian Mathematical Sciences Society}, volume = {35}, year = {2012}, pages = {885-896}, author = {Fernandes, V{\'\i}tor H. and Quinteiro, Teresa M.} } @article {475, title = {The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain}, journal = {Publicationes Mathematicae Debrecen}, volume = {81}, year = {2012}, pages = {11-29}, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J.} } @article {473, title = {Automorphisms of partial endomorphism semigroups}, journal = {Publicationes Mathematicae Debrecen}, volume = {79}, year = {2011}, pages = {23-39}, author = {Ara{\'u}jo, Jo{\~a}o and Fernandes, V{\'\i}tor H. and Jesus, Manuel M. and Maltcev, Victor and Mitchell, James D.} } @article {1208.20057, title = {The cardinal and the idempotent number of various monoids of transformations on a finite chain.}, journal = {Bulletin of the Malaysian Mathematical Sciences Society}, volume = {34}, number = {1}, year = {2011}, pages = {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.

}, keywords = {monoids of transformations, numbers of idempotents, numbers of order-preserving transformations, numbers of orientation preserving transformations}, author = {Fernandes, V{\'\i}tor H. and Gomes, Gracinda M. S. and Jesus, Manuel M.} } @article {471, title = {On the monoids of transformations that preserve the order and a uniform partition}, journal = {Communications in Algebra}, volume = {39}, year = {2011}, pages = {2798{\textendash}2815}, author = {Fernandes, V{\'\i}tor H. and Quinteiro, Teresa M.} } @article {470, title = {Bilateral semidirect product decompositions of transformation monoids.}, journal = {Semigroup Forum}, volume = {82}, number = {2}, year = {2011}, pages = {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.}, keywords = {bilateral semidirect products, free monoids, presentations, transformation semigroups}, doi = {10.1007/s00233-011-9290-1}, author = {Fernandes, V{\'\i}tor H. and Quinteiro, Teresa M.} } @article {MR2678716, title = {Endomorphisms of the semigroup of order-preserving mappings}, journal = {Semigroup Forum}, volume = {81}, number = {2}, year = {2010}, pages = {277{\textendash}285}, issn = {0037-1912}, doi = {10.1007/s00233-010-9220-7}, url = {http://dx.doi.org/10.1007/s00233-010-9220-7}, author = {Fernandes, V{\'\i}tor H. and Jesus, M. M. and Maltcev, V. and Mitchell, J. D.} } @article {MR2735668, title = {On divisors of semigroups of order-preserving mappings of a finite chain}, journal = {Semigroup Forum}, volume = {81}, number = {3}, year = {2010}, pages = {551{\textendash}554}, issn = {0037-1912}, doi = {10.1007/s00233-010-9257-7}, url = {http://dx.doi.org/10.1007/s00233-010-9257-7}, author = {Fernandes, V{\'\i}tor H. and Volkov, M. V.} } @article {MR2488550, title = {Congruences on monoids of transformations preserving the orientation of a finite chain}, journal = {J. Algebra}, volume = {321}, number = {3}, year = {2009}, pages = {743{\textendash}757}, issn = {0021-8693}, doi = {10.1016/j.jalgebra.2008.11.005}, url = {http://dx.doi.org/10.1016/j.jalgebra.2008.11.005}, author = {Fernandes, V{\'\i}tor H. and Gomes, Gracinda M. S. and Jesus, Manuel M.} } @article {MR2395204, title = {The idempotent-separating degree of a block-group}, journal = {Semigroup Forum}, volume = {76}, number = {3}, year = {2008}, pages = {579{\textendash}583}, issn = {0037-1912}, doi = {10.1007/s00233-008-9043-y}, url = {http://dx.doi.org/10.1007/s00233-008-9043-y}, author = {Fernandes, V{\'\i}tor H.} } @article {MR2417624, title = {Normally ordered semigroups}, journal = {Glasg. Math. J.}, volume = {50}, number = {2}, year = {2008}, pages = {325{\textendash}333}, issn = {0017-0895}, doi = {10.1017/S0017089508004230}, url = {http://dx.doi.org/10.1017/S0017089508004230}, author = {Fernandes, V{\'\i}tor H.} } @article {MR2451991, title = {On divisors of pseudovarieties generated by some classes of full transformation semigroups}, journal = {Algebra Colloq.}, volume = {15}, number = {4}, year = {2008}, pages = {581{\textendash}588}, issn = {1005-3867}, author = {Fernandes, V{\'\i}tor H.} } @proceedings {MR2367372, title = {Semigroups and formal languages}, journal = {Proceedings of the International Conference held at the Universidade de Lisboa, Lisboa, July 12{\textendash}15, 2005}, year = {2007}, note = {In honour of the 65th birthday of Donald B. McAlister}, pages = {x+277}, publisher = {World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ}, isbn = {978-981-270-738-3; 981-270-738-7}, editor = {Andr{\'e}, Jorge M. and Fernandes, V{\'{\i}}tor H. and Branco, M{\'a}rio J. J. and Gomes, Gracinda M. S. and Fountain, John and Meakin, John C.} } @article {MR2360515, title = {Largest 2-generated subsemigroups of the symmetric inverse semigroup}, journal = {Proc. Edinb. Math. Soc. (2)}, volume = {50}, number = {3}, year = {2007}, pages = {551{\textendash}561}, issn = {0013-0915}, doi = {10.1017/S0013091505001598}, url = {http://dx.doi.org/10.1017/S0013091505001598}, author = {Andr{\'e}, J. M. and Fernandes, V. H. and Mitchell, J. D.} } @article {MR2230648, title = {Relative abelian kernels of some classes of transformation monoids}, journal = {Bull. Austral. Math. Soc.}, volume = {73}, number = {3}, year = {2006}, pages = {375{\textendash}404}, issn = {0004-9727}, doi = {10.1017/S0004972700035413}, url = {http://dx.doi.org/10.1017/S0004972700035413}, author = {Cordeiro, E. and Delgado, M. and Fernandes, V. H.} } @article {MR2203510, title = {Congruences on monoids of order-preserving or order-reversing transformations on a finite chain}, journal = {Glasg. Math. J.}, volume = {47}, number = {2}, year = {2005}, pages = {413{\textendash}424}, issn = {0017-0895}, doi = {10.1017/S0017089505002648}, url = {http://dx.doi.org/10.1017/S0017089505002648}, author = {Fernandes, V{\'\i}tor H. and Gomes, Gracinda M. S. and Jesus, Manuel M.} } @article {MR2124347, title = {Presentations for some monoids of partial transformations on a finite chain}, journal = {Comm. Algebra}, volume = {33}, number = {2}, year = {2005}, pages = {587{\textendash}604}, issn = {0092-7872}, doi = {10.1081/AGB-200047446}, url = {http://dx.doi.org/10.1081/AGB-200047446}, author = {Fernandes, V{\'\i}tor H. and Gomes, Gracinda M. S. and Jesus, Manuel M.} } @article {1083.20047, title = {Solvable monoids with commuting idempotents}, journal = {Int. J. Algebra Comput.}, volume = {15}, number = {3}, year = {2005}, note = {

n/a

}, pages = {547-570}, abstract = {

The notion of the Abelian kernel of a finite monoid is a generalization of that of the derived subgroup of a finite group. A monoid $M$ is then called solvable if its chain of Abelian kernels terminates with the submonoid of $M$ generated by its idempotents. The main result of this paper is that the finite idempotent commuting monoids bearing this property are precisely those whose subgroups are solvable. In particular any finite aperiodic monoid is Abelian-solvable in this sense. A generalization of the main result of this paper has been published [in Int. J. Algebra Comput. 14, No. 5-6, 655-665 (2004; Zbl 1081.20067)] by the authors and {\i}t S. Margolis and {\i}t B. Steinberg.

}, keywords = {Maltsev products, pseudo-varieties of aperiodic semigroups}, pseudo-varieties of groups, solvable finite semigroups, solvable groups, {Abelian kernels}, doi = {10.1142/S0218196705002414}, author = {Delgado, Manuel and Fernandes, V{\'\i}tor H.} } @article {MR2050894, title = {Abelian kernels of monoids of order-preserving maps and of some of its extensions}, journal = {Semigroup Forum}, volume = {68}, number = {3}, year = {2004}, pages = {335{\textendash}356}, issn = {0037-1912}, doi = {10.1007/s00233-003-0014-z}, url = {http://dx.doi.org/10.1007/s00233-003-0014-z}, author = {Delgado, Manuel and Fernandes, V{\'\i}tor H.} } @inbook {MR2170754, title = {Abelian kernels, solvable monoids and the abelian kernel length of a finite monoid}, booktitle = {Semigroups and languages}, year = {2004}, pages = {68{\textendash}85}, publisher = {World Sci. Publ., River Edge, NJ}, organization = {World Sci. Publ., River Edge, NJ}, doi = {10.1142/9789812702616_0005}, url = {http://dx.doi.org/10.1142/9789812702616_0005}, author = {Delgado, Manuel and Fernandes, V{\'\i}tor H.} } @article {MR2115137, title = {Presentations for some monoids of injective partial transformations on a finite chain}, journal = {Southeast Asian Bull. Math.}, volume = {28}, number = {5}, year = {2004}, pages = {903{\textendash}918}, issn = {0129-2021}, author = {Fernandes, V{\'\i}tor H. and Gomes, Gracinda M. S. and Jesus, Manuel M.} } @proceedings {MR2164866, title = {Semigroups and languages}, journal = {Proceedings of the workshop held at the University of Lisbon, Lisboa, November 27{\textendash}29, 2002}, year = {2004}, pages = {viii+249}, publisher = {World Scientific Publishing Co. Inc.}, address = {River Edge, NJ}, isbn = {981-238-917-2}, editor = {Ara{\'u}jo, Isabel M. and Branco, M{\'a}rio J. J. and Fernandes, V{\'{\i}}tor H. and Gomes, Gracinda M. S.} } @article {MR2104773, title = {On semigroups whose idempotent-generated subsemigroup is aperiodic}, journal = {Internat. J. Algebra Comput.}, volume = {14}, number = {5-6}, year = {2004}, note = {International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes}, pages = {655{\textendash}665}, issn = {0218-1967}, doi = {10.1142/S021819670400202X}, url = {http://dx.doi.org/10.1142/S021819670400202X}, author = {Delgado, Manuel and Fernandes, V{\'{\i}}tor H. and Margolis, Stuart and Steinberg, Benjamin} } @inbook {MR2023797, title = {Presentations for some monoids of partial transformations on a finite chain: a survey}, booktitle = {Semigroups, algorithms, automata and languages (Coimbra, 2001)}, year = {2002}, pages = {363{\textendash}378}, publisher = {World Sci. Publ., River Edge, NJ}, organization = {World Sci. Publ., River Edge, NJ}, doi = {10.1142/9789812776884_0015}, url = {http://dx.doi.org/10.1142/9789812776884_0015}, author = {Fernandes, V{\'\i}tor H.} } @article {MR1945169, title = {A new class of divisors of semigroups of isotone mappings of finite chains}, journal = {Izv. Vyssh. Uchebn. Zaved. Mat.}, number = {3}, year = {2002}, pages = {51{\textendash}59}, issn = {0021-3446}, author = {Fernandesh, V. U.} } @article {MR1842509, title = {A division theorem for the pseudovariety generated by semigroups of orientation preserving transformations on a finite chain}, journal = {Comm. Algebra}, volume = {29}, number = {1}, year = {2001}, pages = {451{\textendash}456}, issn = {0092-7872}, doi = {10.1081/AGB-100000811}, url = {http://dx.doi.org/10.1081/AGB-100000811}, author = {Fernandes, Vitor H.} } @article {MR1831507, title = {The monoid of all injective order preserving partial transformations on a finite chain}, journal = {Semigroup Forum}, volume = {62}, number = {2}, year = {2001}, pages = {178{\textendash}204}, issn = {0037-1912}, author = {Fernandes, V. H.} } @article {MR1830043, title = {On generators and relations for unions of semigroups}, journal = {Semigroup Forum}, volume = {63}, number = {1}, year = {2001}, pages = {49{\textendash}62}, issn = {0037-1912}, author = {Ara{\'u}jo, Isabel M. and Branco, M{\'a}rio J. J. and Fernandes, Vitor H. and Gomes, Gracinda M. S. and Ru{\v s}kuc, N.} } @article {MR1832319, title = {Abelian kernels of some monoids of injective partial transformations and an application}, journal = {Semigroup Forum}, volume = {61}, number = {3}, year = {2000}, pages = {435{\textendash}452}, issn = {0037-1912}, doi = {10.1007/s002330000033}, url = {http://dx.doi.org/10.1007/s002330000033}, author = {Delgado, Manuel and Fernandes, V{\'\i}tor H.} } @article {MR1765325, title = {The monoid of all injective orientation preserving partial transformations on a finite chain}, journal = {Comm. Algebra}, volume = {28}, number = {7}, year = {2000}, pages = {3401{\textendash}3426}, issn = {0092-7872}, doi = {10.1080/00927870008827033}, url = {http://dx.doi.org/10.1080/00927870008827033}, author = {Fernandes, Vitor H.} } @article {MR1611082, title = {Normally ordered inverse semigroups}, journal = {Semigroup Forum}, volume = {56}, number = {3}, year = {1998}, pages = {418{\textendash}433}, issn = {0037-1912}, doi = {10.1007/PL00005955}, url = {http://dx.doi.org/10.1007/PL00005955}, author = {Fernandes, Vitor H.} } @article {MR1428189, title = {Semigroups of order preserving mappings on a finite chain: a new class of divisors}, journal = {Semigroup Forum}, volume = {54}, number = {2}, year = {1997}, pages = {230{\textendash}236}, issn = {0037-1912}, doi = {10.1007/BF02676604}, url = {http://dx.doi.org/10.1007/BF02676604}, author = {Fernandes, Vitor H.} }