Idempotent rank in the endomorphism monoid of a nonuniform partition

Abstract

© 2015 Australian Mathematical Publishing Association Inc. We calculate the rank and idempotent rank of the semigroup ε(X; P) generated by the idempotents of the semigroup T(X;P) which consists of all transformations of the finite set X preserving a nonuniform partition P. We also classify and enumerate the idempotent generating sets of minimal possible size. This extends results of the first two authors in the uniform case.