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Is there a name for regular bands that satisfy $xyx=yx$ for all $x$,$y$?

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A right regular band. The dual notion of left regular bands has been used to study random walks on hyperplane arrangements and oriented matroids. The paper of Ken Brown on Semigroups, rings and Markov chains is a good place to start. My paper with Margolis and Saliola gives more details on the representation theory as does Saliola's solo paper.

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Thanks! By any chance, is there a common name for left regular and right regular bands? – Roman Cheplyaka Jan 13 '13 at 1:37
And one more question if you don't mind: does "band monoid" mean a band which is also a monoid? – Roman Cheplyaka Jan 13 '13 at 2:14
There is no common name for them both. Band monoid means band which is a monoid. – Benjamin Steinberg Jan 13 '13 at 4:14

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