Semigroups uniquely determined by one-sided identity and zero sets

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 $a$. The one-sided identity and zero sets of a semigroup can be utilized in the determination of its maximal subgroups, maximal left and right zero subsemigroups, maximal left and right subgroups, and rectangular band subsemigroups. A band is an idempotent semigroup. Every commutative band is a semilattice and uniquely determined by the left and right identity sets of its elements or equivalently by the left and right zero sets of its elements. We generalize this notion by defining a groupoid or semigroup to be stabilized with respect to binary relations, in particular the binary relations defined by the one-sided identity and zero sets of its elements, if and only if for any groupoid or semigroup on the same set with the same binary relations, their binary operations are identical. We prove every right group with maximal subgroup size $2$ is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements. We define a commutative-rectangular band to be a band in which every pair of elements either commutes or are generalized inverses of each other, and we prove a commutative-rectangular band is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements.