Factoring the Dedekind-Frobenius determinant of a semigroup

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