$\mathcal{J}^{\ast}=\mathcal{D}^{\ast}$ need not hold in finite semigroups

Abstract

We provide an example of minimum size of a finite semigroup with $\mathcal{J}^{\ast}\neq\mathcal{D}^{\ast}$. We introduce the notion of starred stability and prove that every starred stable semigroup has $\mathcal{J}^{\ast}=\mathcal{D}^{\ast}$.