Abstract

Let T(X) denote the full transformation semigroup on a set X. For an arbitrary equivalence relation E on X, we consider a sub semi-group of T(X) defined by: TSE(X)={α∈T(X):(x,xα)∈E for each x∈X} which is called the self-E-preserving transformation semi-group on X. In this paper, we discussed a natural partial order on TSE (X) and characterize when two elements of TSE (X) are related under this order. We also described the left compatibility and right compatibility of each element of TSE (X) with respect to this order.

Key words: Natural partial order, transformation semigroup, compatible.