NATURALLY ORDERED TRANSFORMATION SEMIGROUPS PRESERVING AN EQUIVALENCE

Abstract

Abstract Let 𝒯 X be the full transformation semigroup on a set X and E be a nontrivial equivalence on X . Write then T E ( X ) is a subsemigroup of 𝒯 X . In this paper, we endow T E ( X ) with the so-called natural order and determine when two elements of T E ( X ) are related under this order, then find out elements of T E ( X ) which are compatible with ≤ on T E ( X ). Also, the maximal and minimal elements and the covering elements are described.