Natural partial order on a semi group of self-E-preserving transformations

摘要

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