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- Presentations for monoids of finite partial isometriesPublication . Quinteiro, Teresa; Fernandes, Vítor H.In this paper we give presentations for the monoid DPn of all partial isometries on {1,..., n} and for its submonoid ODPn of all order-preserving partial isometries.
- On semigroups of endomorphisms of a chain with restricted rangePublication . Fernandes, Vitor H.; Honyam, Preeyanuch; Quinteiro, Teresa; Singha, BoorapaLet X be a finite or infinite chain and let be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of and Green's relations on. In fact, more generally, if Y is a nonempty subset of X and is the subsemigroup of of all elements with range contained in Y, we characterize the largest regular subsemigroup of and Green's relations on. Moreover for finite chains, we determine when two semigroups of the type are isomorphic and calculate their ranks.
- On Semigroups of Orientation-preserving Transformations with Restricted RangePublication . Fernandes, Vitor H.; Honyam, Preeyanuch; Quinteiro, Teresa; Singha, BoorapaLet Xn be a chain with n elements (n ∈ ℕ), and let 𝒪𝒫n be the monoid of all orientation-preserving transformations of Xn. In this article, for any nonempty subset Y of Xn, we consider the subsemigroup 𝒪𝒫n(Y) of 𝒪𝒫n of all transformations with range contained in Y: We describe the largest regular subsemigroup of 𝒪𝒫n(Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type 𝒪𝒫n(Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid 𝒪ℛn of all either orientation-preserving or orientation-reversing transformations of Xn.
- Ranks of monoids of endomorphisms of a finite undirected pathPublication . Dimitrova, Ilinka; Fernandes, Vítor Hugo; Koppitz, J.; Quinteiro, TeresaIn 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 wEndPn and EndPn of all weak endomorphisms and all endomorphisms of the undirected path Pn with n vertices. We also consider strong and strong weak endomorphisms of Pn.
- On the monoids of transformations that preserve the order and a uniform partitionPublication . Fernandes, Vítor H.; Quinteiro, TeresaIn this article we consider the monoid O(mxn) of all order-preserving full transformations on a chain with mn elements that preserve a uniformm-partition and its submonoids O(mxn)(+) and O(mxn)(-) of all extensive transformations and of all co-extensive transformations, respectively. We determine their ranks and construct a bilateral semidirect product decomposition of O(mxn) in terms of O(mxn)(-) and O(mxn)(+).
- Ranks and presentations of some normally ordered inverse semigroupsPublication . Caneco, Rita; Fernandes, Vítor Hugo; Quinteiro, TeresaIn 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 N O of all normally ordered (finite) inverse semigroups.
- Partial automorphisms and injective partial endomorphisms of a finite undirected pathPublication . Dimitrova, Ilinka; Fernandes, V. H.; Koppitz, J.; Quinteiro, TeresaIn 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 themonoids IEnd(P-n) and PAut(P-n) of all injective partial endomorphisms and of all partial automorphisms of the undirected path Pn with n vertices. We also describe Green's relations of PAut(P-n) and IEnd(P-n) and calculate their cardinals.
- A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutationsPublication . Fernandes, Vítor Hugo; Quinteiro, TeresaIn this note we consider the monoid PODIn of all monotone partial permutations on {1,....,n} and its submonoids DPn, POIn and ODPn of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids POIn and ODPn are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that PODIn is a quotient of a semidirect product of POIn and the group C-2 of order two and, analogously, DPn is a quotient of a semidirect product of ODPn and C-2.
- On the ranks of certain monoids of transformations that preserve a uniform partitionPublication . Fernandes, Vítor H.; Quinteiro, TeresaThe rank of a semigroup, an important and relevant concept in Semigroup Theory, is the cardinality of a least-size generating set. Semigroups of transformations that preserve or reverse the order or the orientation as well as semigroups of transformations preserving an equivalence relation have been widely studied over the past decades by many authors. The purpose of this article is to compute the ranks of the monoid OR mxn of all orientation-preserving or orientation-reversing full transformations on a chain with mn elements that preserve a uniform m-partition and of its submonoids OP mxn of all orientation-preserving transformations and OD mxn of all order-preserving or order-reversing full transformations. These three monoids are natural extensions of O mxn, the monoid of all order-preserving full transformations on a chain with mnelements that preserve a uniform m-partition.
- Bilateral semidirect product decompositions of transformation monoidsPublication . Fernandes, Vítor H.; Quinteiro, TeresaIn this paper we consider the monoid OR(n) of all full transformations on a chain with n elements that preserve or reverse the orientation, as well as its submonoids OD(n) of all order-preserving or order-reversing elements, OP(n) of all orientation-preserving elements and 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 semidirectproduct of two of its remarkable submonoids.