CAYLEY-SYMMETRIC SEMIGROUPS
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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 Sα is Cayley-symmetric if and only if each Sα is Cayley- symmetric. This enables us to present more Cayley-symmetric semi- groups, which would be non-regular. This result extends the main re- sult 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.
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 [19] is resolved generally. Furthermore, it is proved that a strong semilattice of self-decomposable semigroups Sα is Cayley-symmetric if and only if each Sα is Cayley- symmetric. This enables us to present more Cayley-symmetric semi- groups, which would be non-regular. This result extends the main re- sult 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.
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