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  • Ranks and presentations of some normally ordered inverse semigroups
    Publication . Caneco, Rita; Fernandes, Vítor Hugo; Quinteiro, Teresa
    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 N O of all normally ordered (finite) inverse semigroups.
  • A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutations
    Publication . Fernandes, Vítor Hugo; Quinteiro, Teresa
    In 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.