Cayley-symmetric semigroups
Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 409-419
Published online March 31, 2015
Yongwen Zhu
Yantai University
Abstract : 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
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