Minimal Quasi-Ideals of Generalized Transformation Semigroups
Ronnason Chinram · 2008
Let X and Y be nonempty sets, P (X, Y) denote the set of all map-pings with domain in X and range in Y. By a generalized transformation semigroup of X into Y we mean a semigroup (S(X, Y), θ) where S(X, Y) is a nonempty subset of P (X, Y) and θ ∈ P (Y, X) with αθβ ∈ S(X, Y) for all α, β ∈ S(X, Y) and the operation is ∗ defined by α ∗ β = αθβ for all α, β ∈ S(X, Y). A nonzero quasi-ideal Q of a semigroup S (with or without zero) is said to be minimal if Q does not properly contain any nonzero quasi-ideal of S. In this paper, all minimal quasi-ideals on some generalized transformation semigroups are characterized. As consequences, all minimal quasi-deals on some standard transformation semigroups are determined.