交换半环上内射性的转置对称性
Transpose Symmetry of Injectivity over Commutative Semirings
浏览论文内容
中文总结 AI 辅助
该研究证明交换半环上矩阵映射的内射性、可消性及满射性与转置操作相关，还得到单位交换半环矩阵的稳定有限性定理，揭示了半环的乘法单位元与满射方阵存在性的关联。
中文摘要 AI 辅助
设R为交换半环（不一定具有乘法单位元），A为Mn(R)中的元素。我们证明：Rn上的映射x→Ax是内射的，当且仅当映射x→A^T x是内射的。等价地，乘法半群Mn(R)的左可消元和右可消元是一致的。该证明将形式行列式展开分为偶部和奇部，且不使用减法、加法消去律、群完备化、逆元或乘法单位元。作为推论，我们得到了单位交换半环上矩阵的稳定有限性定理，还证明了满射性在转置下是不变的。实际上，存在正阶的满射方阵会迫使R具有乘法单位元。
英文摘要
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.
