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 $S_{lpha}$ is Cayley-symmetric if and only if each $S_{lpha}$ 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.
展开▼
机译:介绍了Cayley对称半群的概念,并给出了Cayley对称半群的几个等价条件,从而总体上解决了Zhu cite {Zhu-Cayley-2}提出的开放问题。此外,已经证明,当且仅当每个$ S _ { alpha} $是Cayley对称的时,可自分解半群$ S _ { alpha} $的强半格是Cayley对称的。这使我们能够呈现更多的Cayley对称半群,这将是非规则的。该结果扩展了Wang cite {wang}的主要结果,该结果表明正则半群是且仅当它是Clifford半群时才是Cayley对称的。此外,我们讨论了一个半群或一个$ 0 $半群上的Rees矩阵半群的Cayley对称性。
展开▼