Noncommutative inclusion-exclusion, representations of left regular bands and the Tsetlin library

We find a semigroup Qn, whose category of partial representations contains the representation category Rep(Fn) of the free left regular band Fn. We use this to construct a resolution for the absolute kernel of a representation of Fn, for which the kernel Spn 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 Spn to the number of derangements of order n.