One-sided identity and zero sets in semigroups; maximal subsemigroups of certain types
Julia Maddox
TL;DR
This work extends the traditional notions of one-sided identity and zero to subsets of a semigroup by introducing $lidentity(A)$, $ridentity(A)$, $lzero(A)$, and $rzero(A)$, and uses idempotents to organize maximal subsemigroups. It shows that for each idempotent $e$, the maximal left zero subsemigroup is $LZ(e)=ridentity(e)\cap lzero(e)$ and the maximal right zero subsemigroup is $RZ(e)=lidentity(e)\cap rzero(e)$, with rectangular bands decomposing as $T=T_LT_R$ where $T_L\subset LZ(e)$, $T_R\subset RZ(e)$ and $T_RT_L=\{e\}$. The paper also describes maximal subgroups containing $e$ as $RG(e)=H(e)RZ(e)$ and $LG(e)=LZ(e)H(e)$ with $H(e)=eSe$, unifying the structure of rectangular bands and left/right subgroups through one-sided sets. Overall, the approach provides a cohesive framework linking idempotents, one-sided identities/zeroes, and classical semigroup structures to yield a detailed atlas of maximal subsemigroups.
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.