On left absolutely flat bands

Abstract:

A semigroup $S$ is called (left, right) absolutely flat if all of its (left, right) $S$-sets are flat. Let $S = \cup \{ {S_\gamma }:\gamma \in \Gamma \}$ be the least semilattice decomposition of a band $S$. It is known that if $S$ is left absolutely flat then $S$ is right regular (that is, each ${S_\gamma }$ is right zero). In this paper it is shown that, in addition, whenever $\alpha ,\beta \in \Gamma ,\alpha < \beta$, and $F$ is a finite subset of ${S_\beta } \times {S_\beta }$, there exists $w \in {S_\alpha }$ such that $(wu,wv) \in {\theta _R}(F)$ for all $(u,v) \in F({\theta _R}(F)$ denotes the smallest right congruence on $S$ containing $F$). This condition in fact affords a characterization of left absolute flatness in certain classes of right regular bands (e.g. if $\Gamma$ is a chain, if all chains contained in $\Gamma$ have at most two elements, or if $S$ is right normal).