Transpose Symmetry of Injectivity over Commutative Semirings

Abstract

Let R be a commutative semiring, not necessarily with a multiplicative identity, and let A be an element of Mn(R). We prove that the map x to Ax on Rn is injective if and only if x to AT x is injective. Equivalently, the left- and right-cancellative elements of the multiplicative semigroup Mn(R) coincide. The proof splits formal determinant expansions into their even and odd halves; it uses no subtraction, additive cancellation, group completion, inverse, or multiplicative identity. As consequences we recover the stable-finiteness theorem for matrices over unital commutative semirings. We also prove that surjectivity is invariant under transpose. In fact, the existence of a surjective square matrix of positive size forces R to have a multiplicative identity.