Duality theorems for finite semigroups Original Research Article

If R is a ring graded by the semigroup S, then the smash product ring R # S* can be constructed, and in many situations retains categorical “realization” properties analogous to those of the group-graded case. In particular, if S acts as endomorphisms on the ring A, then skew semigroup rings of the form A * S* and S* * A are graded by S, so we may form the “skew-smash” rings (A * S*) # S* and (S* * A) # S*. On the other hand, S acts as endomorphisms on any ring of the form R # S*, so that the skew semigroup rings (R # S*) * S* and S* * (R # S*) may be produced. In particular we may perform this construction when R itself is a skew semigroup ring of the form A * S* or S* * A, thereby yielding “skew-smash-skew” rings. In this article we analyze the resulting skew-smash and skew-smash-skew rings, and prove that each can be realized as a skew semigroup ring for an appropriate ring and (possibly new) semigroup. Inherent in our investigation is the description of a number of methods by which given semigroups can be used to produce new, related semigroups. As one consequence of our results we provide a broader context for some of the group-theoretic “duality” results of Cohen and Montgomery.