A note on an alternating upper bound for random walks on semigroups

We consider random walks on idempotent semigroups, called Left Regular Bands, satisfying the relation xyx=xyxyx=xy for any two elements xx and yy of the semigroup. We give an alternating upper bound for the total variation distance of a random walk on a Left Regular Band semigroup, improving the previous bound by Brown and Diaconis.