Noncommutative Inclusion-Exclusion, Representations Of Left Regular Bands And The Tsetlin Library

We find a semigroup Q(n), whose category of partial representations contains the representation category Rep(F-n) of the free left regular band F-n. We use this to construct a resolution for the absolute kernel of a representation of F-n, for which the kernel Sp(n) of the Markov operation in the Tsetlin library model is a prominent example. We obtain a formula for the dimension of the absolute kernel, generalizing the equality of the dimension of Sp(n) to the number of derangements of order n.