Bulletin of the
Korean Mathematical Society BKMS

The concept of Cayley-symmetric semigroups is introduced, and several equivalent conditions of a Cayley-symmetric semigroup are given so that an open problem proposed by Zhu \cite{Zhu-Cayley-2} is resolved generally. Furthermore, it is proved that a strong semilattice of self-decomposable semigroups $S_{\alpha}$ is Cayley-symmetric if and only if each $S_{\alpha}$ is Cayley-symmetric. This enables us to present more Cayley-symmetric semigroups, which would be non-regular. This result extends the main result of Wang \cite{wang}, which stated that a regular semigroup is Cayley-symmetric if and only if it is a Clifford semigroup. In addition, we discuss Cayley-symmetry of Rees matrix semigroups over a semigroup or over a $0$-semigroup.

Keywords: generalized Cayley graph, Cayley-symmetric semigroup, strong semilattice of semigroups, self-decomposable

MSC numbers: 05C25, 20M12, 20M17