Semigroups of Partial Transformations Preserving an Equivalence Relation
Pei Huisheng, Huijuan Zhou · Advances in Mathematics · 2009
Let X be a set with |X|≥3,P_X the semigroup of all partial transformations on the set X.Let E be an equivalence relation on X and define P_E(X)={f∈P_X:■x,y∈domf,(x,y)∈E■(f(x),f(y))∈E}. Then P_E(X)forms a subsemigroup of P_X.In this paper,Green's equivalences and regularity of the semigroup P_E(X)are investigated and the condition under which P_E(X)is abundant is observed.