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

TL;DR

The paper demonstrates that in finite semigroups, the equality of the Green's relations $\,\mathcal{J}^*\,$ and $\,\mathcal{D}^*\,$ does not always hold, providing a minimal counterexample and establishing conditions for equality.

Contribution

It introduces the concept of starred stability and proves that starred stable semigroups always satisfy $\,\mathcal{J}^* = \mathcal{D}^*\,$, clarifying the relationship between these relations.

Findings

Constructed the smallest finite semigroup with $\,\mathcal{J}^* \neq \mathcal{D}^*$.

Defined starred stability and proved it implies $\,\mathcal{J}^* = \mathcal{D}^*\,$.

Provided insight into the structure of finite semigroups and Green's relations.

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 $\mathcal{J}^{\ast}=\mathcal{D}^{\ast}$.