关于序超半群的左超理想的研究

An Investigation on Left Hyperideals of Ordered Semihypergroups

中文关键词:  序超半群  极小左超理想  极大左超理想  弱素左超理想  拟素左超理想  弱拟素左超理想

英文关键词:ordered semihypergroup  minimal left hyperideal  maximal left hyperideal  weakly prime left hyperideal  quasi-prime left hyperideal  weakly quasi-prime left hyperideal

中文摘要:

      本文在序超半群中引入了极小左超理想和极大左超理想的概念, 并讨论了它们的一些相关性质. 进一步地, 引入了序超半群的弱素左超理想、拟素左超理想、拟半素左超理想及弱拟素左超理想的概念, 并讨论这四种素超理想之间的关系. 而且通过左超理想和弱~$m$-系刻画了序超半群的弱拟素左超理想. 同时, 借助于$m$-系对序超半群的拟素左超理想给出刻画. 尤其证明了序超半群$S$是强半单的当且仅当$S$的每个左超理想是$S$的包含它的所有拟素左超理想的交.

英文摘要:

      In this paper, the concepts of minimal and maximal left hyperideals in ordered semihypergroups are introduced, and several related properties are investigated. Furthermore, we introduce the concepts of weakly prime, quasi-prime, quasi-semiprime and weakly quasi-prime left hyperideals of an ordered semihypergroup, and establish the relationship among the four classes of left hyperideals. Moreover, we give some characterizations of weakly quasi-prime left hyperideals by the left hyperideals and weakly $m$-systems. We also characterize the quasi-prime left hyperideals in terms of the $m$-systems. In particular, we prove that an ordered semihypergroup $S$ is strongly semisimple if and only if every left hyperideal of $S$ is the intersection of all quasi-prime left hyperideals of $S$ containing it.