AI 中文总结
本文在$\text{Z}_2$-分次框架下引入多种Hom型反代数超结构，研究其构造、性质及算子作用，建立结构间关系，为非交换Poisson型代数的推广提供统一研究框架。
AI 中文摘要
本文在$\boldsymbol{\text{Z}_2}$-分次框架下发展了多种Hom型反代数结构的理论，引入并研究了Hom-反结合超代数、Hom-反dendriform超代数和Hom-反pre-Lie超代数，这些结构是对应反代数结构的Hom型推广。文中提供了多种构造与例子，同时探讨了结构性质及表示论方面的内容，特别研究了反超$\boldsymbol{\text{O}}$-算子和反Rota-Baxter算子在Hom-反dendriform超代数构造中的作用。此外，本文还引入非交换Hom-pre-Poisson超代数和非交换Hom-anti-pre-Poisson超代数的概念，它们是分次框架下非交换Poisson型结构的Hom型扩展，这些结构通过合适的相容性条件自然结合了Hom-反pre-Lie超代数与Hom-反dendriform超代数，文中建立了所引入结构间的多种关系，为研究非交换Poisson型代数的扭曲与分次推广提供了统一框架。
英文摘要
In this paper, we develop the theory of several Hom-type anti-algebraic structures in the $\mathbb{Z}_2$-graded setting. We introduce and study Hom-anti-associative superalgebras, Hom-anti-dendriform superalgebras, and Hom-anti-pre-Lie superalgebras, which arise as Hom-type generalizations of the corresponding anti-algebraic structures. Various constructions and examples are provided, together with structural properties and representation-theoretic aspects. In particular, we investigate the role of anti-super-$\mathcal{O}$-operators and anti-Rota-Baxter operators in the construction of Hom-anti-dendriform superalgebras. Furthermore, we introduce the notion of noncommutative Hom-pre-Poisson superalgebras and noncommutative Hom-anti-pre-Poisson superalgebras as Hom-type extensions of noncommutative Poisson-type structures in the graded framework. These structures naturally combine Hom-anti-pre-Lie and Hom-anti-dendriform superalgebras through suitable compatibility conditions. Several relationships between the introduced structures are established, providing a unified framework for studying twisted and graded generalizations of noncommutative Poisson-type algebras.
