Naturally Ordered Semigroups of Partial Transformations Preserving an Equivalence Relation
Pei Huisheng, Deng Weina · Communications in Algebra · 2013
Let X be a set with |X| ≥3, P(X) the semigroup of all partial transformations on the set X, E be an equivalence relation on X, and Then P E (X) forms a subsemigroup of P(X). In this article, we consider the natural order ≤on P E (X) and determine when two elements of P E (X) are related under this order. Then we characterize those elements of P E (X) which are left (right) compatible with ≤on P E (X). Besides, the minimal (maximal) elements are described.