Abstract

A semigroup is called a Hua semigroup if any mapping $h:S\to S$ satisfying $$(\forall a,b\in S)h(ab)=h(a)h(b)\mathrm{}\text{or}\mathrm{}h(b)h(a)$$ is either a homomorphism or an anti-homomorphism. We use this name for such a semigroup since it is relevant to a well-known result in ring theory, which is established by L. K. Hua in 1949. C. M. Reis and H. J. Shyr first translated Hua’s result from rings to semigroups in 1977. In this paper, elementary characterizations, a hierarchy and closure properties of the class of Hua semigroups and its four subclasses are given. We show that every cancellative semigroup is a Hua semigroup, which generalizes Reis and Shyr’s result.

摘要

有映射的半群称为华半群 $H：秒\to 秒$ 满意 $$(\forall \mathrm{一种},乙\in 秒)H(\mathrm{一种}乙)=H(\mathrm{一种})H(乙)\mathrm{}\text{或者}\mathrm{}H(乙)H(\mathrm{一种})$$是同态或反同态。我们对这样的半群使用这个名称，因为它与环理论中的一个著名结果有关，该结果由 LK Hua 在 1949 年建立。CM Reis 和 HJ Shyr 于 1977 年首次将华的结果从环翻译成半群。在这篇论文中, 初等刻画, 华半群类及其四个子类的层次和闭包性质。我们证明了每个取消半群都是一个 Hua 半群，它推广了 Reis 和 Shyr 的结果。