A problem on generalized Cayley graphs of semigroups

Zhu (Semigroup Forum 84(3), 144–156, 2012) investigated some combinatorial properties of generalized Cayley graphs of semigroups. In Remark 3.8 of (Zhu, Semigroup Forum 84(3), 144–156, 2012), Zhu proposed the following question: It may be interesting to characterize semigroups S such that Cay(S,ω _l )=Cay(S,ω _r ). In this short note, we prove that for any regular semigroup S, Cay(S,ω _l )=Cay(S,ω _r ) if and only if S is a Clifford semigroup.