LR-正则Clifford半环的性质和结构

THE PROPERTY AND STRUCTURE OF LR-NORMAL CLIFFORD SEMIRINGS

为了完善半环的Clifford层次的研究,基于已有研究结果,利用LR-正则带,定义了LR-正则Clifford半环.一个半环S称为LR-正则Clifford半环,若S是矩形环Sα的分配格D(α∈D),并且E+(S)是一个LR-正则带.这类半环是左(右)Clifford半环的真推广.利用半群的研究方法,借助LR-正则纯正群并半群、矩形Clifford半环和拟Clifford半环的结构,给出了半环是LR-正则Clifford半环的充分必要条件,最后在LR-正则Clifford半群织积结构的前提下,给出了在一定条件下LR-正则Clifford半环的织积结构.所得结果推广了左(右)Clifford半环的相应结论.

In order to improve the research of Clifford hierarchy of semirings,based on the existing research results,we define LR-regular Clifford semirings by using LR-regular band.A semiring S is said to be an LR-regular Clifford semiring if S is a distributive lattice of rectanglar ring Sα(α∈D)and E+(S)is an LR-regular band.Such semirings are the proper generalization of left(right)Clifford semirings.By using the research method of semigroups,with the help of the structure of LR-regular orthogroups,rectangular Clifford semirings and quasi Clifford semigroups,the necessary and sufficient conditions for the semirings to be LR-regular Clifford semirings are given,and the spined product structure of LR-regular Clifford semirings under certain conditions is given.The results generalize the conclusions of left(right)Clifford semirings.