One-sided identity and zero sets in semigroups; maximal subsemigroups of certain types

Abstract: Given semigroup $S$ and nonempty $A \subset S$, $lidentity(A)$ is the set of $b \in S$ with $ba=a$ for all $a \in A$; $lzero(A)$ is the set of $b \in S$ with $ba=b$ for all $a \in A$. There are similar definitions for $ridentity(A)$ and $rzero(A)$. A one-sided identity or zero is an idempotent, and an idempotent is a one-sided identity or zero for some subsemigroup. Every idempotent of a semigroup exists in a maximal left [right] zero subsemigroup and in a maximal right [left] subgroup. We can also describe all rectangular band subsemigroups containing that idempotent. These subsemigroups are constructed using one-sided identity and zero sets. This methodology also provides a new approach to the known structure of rectangular bands and left and right groups.