Prime objects among idempotent semigroups

Determine whether the class of all idempotent semigroups (semigroups satisfying the identity xx = x) contains any prime objects in Tarski’s sense.

Background

Section 5 reviews four questions from McKenzie, McNulty and Taylor [6], noting that one has been answered but two remain open. One of these concerns idempotent semigroups, a well-studied class characterized by the identity xx = x.

The authors explicitly state that, to their knowledge, this question remains open, and present it verbatim from [6].

References

So far as I know, the other two questions remain open. Question 5.1. [6, p.263] Is there any prime in the class of all idempotent semigroups?

— On semigroups that are prime in the sense of Tarski, and groups prime in the senses of Tarski and of Rhodes (2409.15541 - Bergman, 23 Sep 2024) in Question 5.1, Section 5