AI 中文总结
本文将Steinberg行列式理论扩展至有限域扭曲收缩半群代数，建立其弗罗贝尼乌斯判据，构造该类代数的齐次权重，刻画满足平均性质的主理想，拓展了齐次权重的适用范围。
AI 中文摘要
有限半群的行列式与其对应半群代数的弗罗贝尼乌斯性质密切相关，在编码理论应用中具有重要意义。本文研究有限域上扭曲收缩半群代数的行列式，将Steinberg的行列式理论扩展至该情形，建立有限域上扭曲收缩半群代数的弗罗贝尼乌斯判据；还发展了双生成扭曲收缩幺半群代数的结构理论，特别确定了一类矩阵的行列式仅依赖其零-非零模式、与上同调群取值无关的代数，因此只需验证平凡特征的行列式条件即可推出所有特征的对应结果，在适当假设下这类半群由少量定义恒等式完全确定。受有限弗罗贝尼乌斯环上齐次权重的特征论构造启发，本文为弗罗贝尼乌斯扭曲收缩半群代数构造齐次权重，并研究非弗罗贝尼乌斯情形，特别刻画了特征平均权重满足齐次权重平均性质的主理想，从而将齐次权重的经典概念扩展至指定的主理想族。
英文摘要
The determinant of a finite semigroup is closely related to the Frobenius property of its corresponding semigroup algebra, and it plays a significant role in coding theory applications. In this paper, we study determinants of twisted contracted semigroup algebras over finite fields. We extend Steinberg's determinant theory to this setting by establishing a Frobenius criterion for twisted contracted semigroup algebras over finite fields. We also develop a structural theory for two-generated twisted contracted monoid algebras. In particular, we identify classes for which the determinant of the associated matrix depends only on its zero--nonzero pattern, and is therefore independent of the values of the cocycle. Consequently, it suffices to verify the determinant condition for the trivial character, from which the corresponding result for all characters follows. Under suitable assumptions, these semigroups are completely determined by a small set of defining identities. Motivated by the character-theoretic construction of homogeneous weights on finite Frobenius rings, we construct homogeneous weights for Frobenius twisted contracted semigroup algebras and investigate the non-Frobenius case. In particular, we characterize the principal ideals on which the character-average weight satisfies the averaging property of homogeneous weights and thereby extend the classical notion of homogeneous weight to a prescribed family of principal ideals.
