  The representation theory of finite groups began with Frobenius's
factorization of Dedekind's group determinant. In this paper, we consider the
case of the semigroup determinant. The semigroup determinant is nonzero if and
only if the complex semigroup algebra is Frobenius, and so our results include
applications to the study of Frobenius semigroup algebras. We explicitly factor
the semigroup determinant for commutative semigroups and inverse semigroups. We
recover the Wilf-Lindstr\"om factorization of the semigroup determinant of a
meet semilattice and Wood's factorization for a finite commutative chain ring.
The former was motivated by combinatorics and the latter by coding theory over
finite rings. We prove that the algebra of the multiplicative semigroup of a
finite Frobenius ring is Frobenius over any field whose characteristic doesn't
divide that of the ring. As a consequence we obtain an easier proof of
Kov\'acs's theorem that the algebra of the monoid of matrices over a finite
field is a direct product of matrix algebras over group algebras of general
linear groups (outside of the characteristic of the finite field).
