@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 {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 {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 {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 {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 {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 {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 {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 {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 {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.} }