• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1169-1183
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• 2015

We describe generalized Cayley graphs of rectangular groups, so that we obtain (1) an equivalent condition for two Cayley graphs of a rectangular group to be isomorphic to each other, (2) a necessary and sufficient condition for a generalized Cayley graph of a rectangular group to be (strong) connected, (3) a necessary and sufficient condition for the colour-preserving automorphism group of such a graph to be vertex-transitive, and (4) a sufficient condition for the automorphism group of such a graph to be vertex-transitive.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.409-419
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• 2015

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 [19] 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 semi-groups, which would be non-regular. This result extends the main result of Wang [14], 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.