三类半环上矩阵的正交性

The orthogonality of matrices over three kinds of semirings

摘要

对于n×n阶矩阵A和B,若矩阵AB和BA都为零矩阵,则称A和B正交。若A ²为零矩阵,则称A为自正交。本文研究一类特殊tropical(0,-1)矩阵的正交性以及二元布尔代数和链半环上矩阵的自正交性。研究二元布尔代数上矩阵的自正交性,间接刻画二元布尔代数上的零方矩阵形式。

Abstract

For n×n matrices A and B, they are considered orthogonal when both AB and BA are zero matrices, and A is deemed self-orthogonal when A ² is a zero matrix. The orthogonality of a specific class of tropical (0,-1) matrices, as well as the self-orthogonality of matrices on binary Boolean algebras and chain semirings is studied. The self-orthogonality of matrices on binary Boolean algebras are studied and the zero-square matrix form on those algebras is indirectly characterized.

关键词

Tropical代数 / 正交性 / 二元布尔代数 / 链半环

Key words

tropical algebra / orthogonality / binary Boolean algebra / chain semiring

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基金资助

国家自然科学基金资助项目(12371024)

重庆市教委科学技术研究项目(KJZD-K202401102)

重庆市自然科学基金创新发展联合基金(CSTB2025NSCQ-LZX0067)