Notes on a problem on weakly exponential $\Delta$-semigroups

Abstract

A semigroup $S$ is called a weakly exponential semigroup if, for every couple $(a,b)\in S\times S$ and every positive integer $n$, there is a non-negative integer $m$ such that $(ab)^{n+m}=a^nb^n(ab)^m=(ab)^ma^nb^n$. A semigroup $S$ is called a $\Delta$-semigroup if the lattice of all congruences of $S$ is a chain with respect to inclusion. The weakly exponential $\Delta$-semigroups were described in [5]: A. Nagy, Weakly exponential $\Delta$-semigroups, Semigroup Forum, 40(1990), 297-313. Although the existence of two types of them (T2R and T2L semigroups) is an open question, Theorem 3.11 of [5] gives necessary and sufficient conditions for a semigroup to be a T2R [T2L] semigroup. In our present paper we give a little correction of condition (v) of Theorem 3.11 of [5], and prove some new results which are addendum to the problem: Doest there exist a T2R [T2L] semigroup?