Completely 0-simple semigroup with the basis property

Abstract

A semigroup [Formula: see text] is said to have the Basis Property if for any subsemigroup [Formula: see text] of a semigroup [Formula: see text], any two bases for [Formula: see text] have the same cardinality. The structure of completely [Formula: see text]-simple semigroup with the Basis Property is described. In particular, we proved that each completely [Formula: see text]-simple semigroup [Formula: see text] has the Basis Property if and only if [Formula: see text] satisfies one of the following conditions: (1) [Formula: see text] is produced from a completely simple semigroup with adjoint zero. (2) [Formula: see text] is an isomorphic to Rees’s semigroup [Formula: see text] over a group [Formula: see text] with sandwich matrix [Formula: see text] such that [Formula: see text], [Formula: see text], in addition [Formula: see text] has a zero in every row and column.