Naturally Ordered Transformation Semigroups Preserving an Equivalence Relation and a Cross-section
Lei Sun, Weina Deng, Pei Huisheng · Algebra Colloquium · 2011
The paper is concerned with the so-called natural order on the semigroup [Formula: see text], where [Formula: see text] is the full transformation semigroup on a set X, E is a non-trivial equivalence on X and R is a cross-section of the partition X/E induced by E. We determine when two elements of TE(X,R) are related under this order, find elements of TE(X,R) which are compatible with ≤ on TE(X,R), and observe the maximal and minimal elements and the covering elements.