Abstract
Let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation ρ on X, let ρ^ be the restriction of ρ on Y, R a cross-section of Y/ρ^ and define T(X,Y,ρ,R) to be the set of all total transformations α from X into Y such that α preserves both ρ (if (a,b)∈ρ, then (aα,bα)∈ρ) and R (if r∈R, then rα∈R). T(X,Y,ρ,R) is then a subsemigroup of T(X,Y). In this paper, we give descriptions of Green’s relations on T(X,Y,ρ,R), and these results extend the results on T(X,Y) and T(X,ρ,R) when taking ρ to be the identity relation and Y=X, respectively.
Highlights
Let X be a nonempty set and TpXq denote the semigroup containing all full transformations from X into itself with the composition
It is well-known that TpXq is a regular semigroup, as shown in Reference [1]
One of the subsemigroups of TpXq is related to an equivalence relation ρ on X and a cross-section R of the partition X{ρ, namely TpX, ρ, Rq, which was first considered by Araújo and Konieczny in 2003 [2], and is defined by
Summary
Let X be a nonempty set and TpXq denote the semigroup containing all full transformations from X into itself with the composition. One of the subsemigroups of TpXq is related to an equivalence relation ρ on X and a cross-section R of the partition X{ρ (i.e., each ρ-class contains exactly one element of R), namely TpX, ρ, Rq, which was first considered by Araújo and Konieczny in 2003 [2], and is defined by. They studied automorphism groups of centralizers of idempotents They determined Green’s relations and described the regular elements of TpX, ρ, Rq in 2004 [3]. Let Y be a nonempty subset of the set X Consider another subsemigroup of TpXq, which was first introduced by Symons [4] in 1975, called TpX, Yq, defined by.
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