One-sided identity and zero sets and an elementary approach to the Rees-Sushkevich Theorem

Abstract

For a groupoid $S$ with elements $a$ and $b$, if $ba=a$, then $b$ is a left identity of $a$ and $a$ is a right zero of $b$. We define the left identity set of $a$ to be the set of all left identities of $a$ in $S$, and similarly for the right identity set of $a$ in $S$. We defined the left zero set of $a$ to be the set of all left zeroes of $a$ in $S$, and similarly for the right zero set of $a$. We use the one-sided identity and zero sets of a semigroup in the determination of the structure of its maximal subgroups, maximal right and left zero subsemigroups, maximal right and left subgroups, rectangular band subsemigroups, completely simple subsemigroups, rectangular subgroups, and completely $0$-simple subsemigroups. An elementary approach of the Rees-Sushkevich Theorem follows. Then we define rectangular $0$-bands and rectangular $0$-groups as completely $0$-simple analogues of rectangular bands and rectangular groups.