Dimitrova, I., Vítor H. Fernandes, J. Koppitz, and T. M. Quinteiro. "
On monoids of endomorphisms of a cycle graph."
Mathematica Slovaca (DOI 10.1515/ms-2024-0078; Online 15 October 2024). 74.5 (2024): 1071-1088.
AbstractIn 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's relations and regularity of these monoids and calculate their cardinalities.
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.
AbstractIn 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).
Dimitrova, I., Vítor H. Fernandes, and J. Koppitz. "
On partial endomorphisms of a star graph."
Quaestiones Mathematicae (DOI 10.2989/16073606.2024.2374796; Online 31 July 2024). 47.12 (2024): 2485-2505.
AbstractIn 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's relations, calculate their cardinalities and study their regularity.
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).
AbstractIn 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, 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.
Fernandes, Vítor H. "
On the monoid of partial isometries of a wheel graph."
Asian-European Journal of Mathematics (DOI 10.1142/S1793557123502388; Online 16 Dec 2023). 17.1 (2024): 2350238 (18 pages).
AbstractIn 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's relations of DPW_n and of its three considered subsemigroups.
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).
AbstractIn 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.