AI 中文总结
本文用Tsallis q-和替代普通加法律，引入有限集合的公理q-基数、提出非线性矩阵合成，证明相关代数性质，将非广延统计力学与有限代数等领域关联。
AI 中文摘要
Tsallis q-和是非广延统计力学的基本非线性合成律之一。尽管它在连续场景中已被广泛研究，但它对有限数学结构的影响仍较少被探索。本文通过用Tsallis q-和替代普通加法律来研究该问题。我们首先引入满足非线性可加性原理的有限集合的公理q-基数，证明了其存在性与唯一性，得到了当q→1时连续恢复经典基数的显式表达式。随后，我们提出由相同变形诱导的非线性矩阵合成，并确立其主要代数性质，包括结合律、单位元，以及用普通矩阵对易子表征的非交换性。在M₂(ℤ₆)上的示例表明，当1-q为零因子时，该变形可改变有限矩阵代数的中心。这些结果表明，Tsallis q-和为构建非线性有限数学结构提供了自然机制，并将非广延统计力学与有限代数及非线性数学物理学联系起来。
英文摘要
The Tsallis $q$-sum is one of the fundamental nonlinear composition laws of nonextensive statistical mechanics. Although it has been extensively investigated in continuous settings, its implications for finite mathematical structures remain less explored. Here we investigate this question by replacing ordinary additive laws with the Tsallis $q$-sum. We first introduce an axiomatic $q$-cardinality of finite sets satisfying a nonlinear additivity principle. Existence and uniqueness are established, leading to an explicit expression that continuously recovers the classical cardinality as $q\to1$. We then propose a nonlinear matrix composition induced by the same deformation and establish its principal algebraic properties, including associativity, the neutral element, and a characterization of its noncommutativity in terms of the ordinary matrix commutator. An illustrative example over $M_2(\mathbb Z_6)$ shows how the deformation can modify the center of a finite matrix algebra when $1-q$ is a zero divisor. These results indicate that the Tsallis $q$-sum provides a natural mechanism for constructing nonlinear finite mathematical structures and connects nonextensive statistical mechanics with finite algebra and nonlinear mathematical physics.
