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arXiv · 2608.16205

Transpose Symmetry of Injectivity over Commutative Semirings

Sixuan Gu·Wei Qi·Yaoyu Cheng
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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.

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Sixuan Gu, Wei Qi, Yaoyu Cheng. 2026-08-17. Transpose Symmetry of Injectivity over Commutative Semirings. https://arxiv.org/abs/2608.16205

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