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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On left absolutely flat bands


Authors: Sydney Bulman-Fleming and Kenneth McDowell
Journal: Proc. Amer. Math. Soc. 101 (1987), 613-618
MSC: Primary 20M50; Secondary 20M10
DOI: https://doi.org/10.1090/S0002-9939-1987-0911019-X
MathSciNet review: 911019
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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).


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DOI: https://doi.org/10.1090/S0002-9939-1987-0911019-X
Keywords: Right regular band, left absolutely flat semigroup
Article copyright: © Copyright 1987 American Mathematical Society