On the Semigroup of Partial Isometries of a Finite Chain

Combinatorial Results for Certain Semigroups of Order-Decreasing Partial Isometries of a Finite Chain

“…First, notice that from [1, Lemma 2.1] we deduce that number of L-classes in K(n, p) = {α ∈ DP n : h(α) = p} (as well as the number of R-classes there) is n p . To describe the D-classes in DP n and ODP n , first we recall (from [1]) that the gap and reverse gap of the image set of α (with h(α) = p) are ordered (p − 1)-tuples defined as follows:…”

“…Analogous to Al-Kharousi et al [2], this paper investigates the combinatorial properties of DDP n and ODDP n , thereby complementing the results in Kehinde et al [13] which dealt mainly with the algebraic and rank properties of these semigroups. In this section we introduce basic definitions and terminology as well as quote some elementary results from Section 1 of Al-Kharousi et al [1] and Kehinde et al [13] that will be needed in this paper. In Section 2 we obtain the cardinalities of two equivalences defined on ODDP n and DDP n .…”

A Note on Bilateral Semidirect Product Decompositions of Some Monoids of Order-Preserving Partial Permutations

“…Observe that ODP n , POI n , DP n and PODI n are all inverse submonoids of the symmetric inverse monoid (i.e. the monoid of all partial permutations) I n on X n (see [3,14]). Obviously, POI n ⊆ PODI n and ODP n = DP n ∩ POI n and, as observed by Al-Kharousi et al [3], we also have DP n ⊆ PODI n .…”

“…the monoid of all partial permutations) I n on X n (see [3,14]). Obviously, POI n ⊆ PODI n and ODP n = DP n ∩ POI n and, as observed by Al-Kharousi et al [3], we also have DP n ⊆ PODI n . Moreover, it is easy to check that ODP n = {s ∈ I n | is − js = i − j, for i, j ∈ Dom(s)}.…”

Presentations for monoids of finite partial isometries

“…See also [11], for a survey on known presentations of transformations monoids. We notice that the first author together with Delgado [6,7] have computed the abelian kernels of the monoids POI n and PODI n , by using a method that is strongly dependent of given presentations of the monoids.The study of semigroups of finite partial isometries was initiated by 3]. The first of these two papers is dedicated to investigate some combinatorial properties of the monoid DP n of all partial isometries on {1, .…”