Left (Right) Regular and Transposition Regular Semigroups and Their Structures

Regular semigroups and their structures are the most wonderful part of semigroup theory, contents very rich. In order to explore more regular semigroups, this paper extends relevant classical conclusions from a new perspective: by transforming positions elements in regularity conditions, some conditions (collectively referred as transposition regularity) obtained, concepts various introduced (L1/L2/L3, R1/R2/R3-transposition etc.). Their relations with completely left (right) proposed Clifford Preston, analyzed. properties studied aspects idempotents, local identity elements, inverse subsemigroups so on. decomposition theorems proved respectively, necessary sufficient for become obtained.