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2023
Fernandes, Vítor H., and Tânia Paulista. "On the monoid of partial isometries of a cycle graph." Turkish Journal of Mathematics (DOI 10.55730/1300-0098.3460). 47 (2023): 1746-1760. AbstractWebsite

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's relations of DPC_n and calculate its cardinality and rank.

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 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.