On left absolutely flat bands
Sydney Bulman‐Fleming, Kenneth McDowell · Proceedings of the American Mathematical Society · 1987
A semigroup S S is called (left, right) absolutely flat if all of its (left, right) S S -sets are flat. Let S = ∪ { S γ : γ ∈ Γ } S = \cup \{ {S_\gamma }:\gamma \in \Gamma \} be the least semilattice decomposition of a band S S . It is known that if S S is left absolutely flat then S S is right regular (that is, each S γ {S_\gamma } is right zero). In this paper it is shown that, in addition, whenever α , β ∈ Γ , α > β \alpha ,\beta \in \Gamma ,\alpha > \beta , and F F is a finite subset of S β × S β {S_\beta } \times {S_\beta } , there exists w ∈ S α w \in {S_\alpha }