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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 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)(+).
- 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.
- The cardinal of various monoids of transformations that preserve a uniform partitionPublication . Fernandes, Vítor H.; Quinteiro, TeresaIn this paper we give formulas for the number of elements of the monoids ORm x n of all full transformations on it finite chain with tun elements that preserve it uniform m-partition and preserve or reverse the orientation and for its submonoids ODm x n of all order-preserving or order-reversing elements, OPm x n of all orientation-preserving elements, O-m x n of all order-preserving elements, O-m x n(+) of all extensive order-preserving elements and O-m x n(-) of all co-extensive order-preserving elements.