#### Document Type

Article

#### Publication Date

12-1987

#### Department

Mathematics

#### Abstract

A semigroup *S* is called (left, right) absolutely flat if all of its (left, right) *S*-sets are flat. Let *S* = ∪{*S*_{β} : β∈Γ} be the least semillatice decomposition of a band *S*. It is known that if *S* is left absolutely flat then *S* is right regular (that is, each *S*_{β} is a right zero). In this paper it is shown that, in addition, whenever α, β∈Γ, α < β, and *F* is a finite subset of *S*_{β} x *S*_{β}, there exists *w*∈*S*_{α} such that (*wu*, *wv*) ∈θ_{R}(*F*) for all (*u*, *v*)∈*F*(θ_{R}(*F*) denotes the smallest right congruence on *S* containing *F*). This condition in fact afford a characterization of left absolute flatness in certain classes of right regular bands (e.g. if Γ is a chain, if all chains contained in Γ have at most two elements, or if *S* is right normal.

#### Recommended Citation

Bulman-Fleming, Sydney and McDowell, Kenneth, "On Left Absolutely Flat Bands" (1987). *Mathematics Faculty Publications*. 5.

https://scholars.wlu.ca/math_faculty/5

## Comments

This article was originally published in

Proceedings of the American Mathematical Society, 101(4): 613-618. (c) 1987 American Mathematical Society