从克隆到cm-幺半群
From clones to cm-monoids
AI总结：
研究引入克隆合并幺半群统一代数框架，整合多种克隆相关概念并建模无穷运算。建立克隆代数与有限秩cm-幺半群范畴等价性，还发展其模理论，为代数结构和CSP复杂性理论提供新见解。
摘要：
函数克隆在泛代数和理论计算机科学中都起着基础性作用。在这项工作中，我们引入了克隆合并幺半群（cm-幺半群），这是一个统一的单类代数框架，它整合了抽象克隆、克隆代数（由第一和第三作者先前引入）以及诺伊曼的阿列夫零抽象克隆，同时对无穷运算的相互作用进行建模。cm-幺半群将幺半群结构与一种称为合并代数的新代数结构相结合，捕捉了无穷运算序列的基本性质。我们建立了克隆代数与有限秩cm-幺半群之间的范畴等价性。这种等价性通过限制在抽象克隆、有限维克隆代数和有限维、有限秩cm-幺半群之间产生了三重等价性，并且它本身是通过从部分无穷克隆代数（它推广了克隆代数）和外延cm-幺半群之间的范畴等价性限制得到的。在一项配套工作中，我们发展了cm-幺半群上的模理论，为多态性和不变关系提供了一种统一的方法，希望能为代数结构和CSP复杂性理论提供新的见解。
英文摘要：
Clones of functions play a foundational role in both universal algebra and theoretical computer science. In this work, we introduce clone merge monoids (cm-monoids), a unifying one-sorted algebraic framework that integrates abstract clones, clone algebras (previously introduced by the first and the third author), and Neumann's aleph0-abstract clones, while modelling the interplay of infinitary operations. Cm-monoids combine a monoid structure with a new algebraic structure called merge algebra, capturing essential properties of infinite sequences of operations. We establish a categorical equivalence between clone algebras and finitely-ranked cm-monoids. This equivalence yields by restriction a three-fold equivalence between abstract clones, finite-dimensional clone algebras, and finite-dimensional, finitely ranked cm-monoids, and is itself obtained by restriction from a categorical equivalence between partial infinitary clone algebras (which generalise clone algebras) and extensional cm-monoids. In a companion work, we develop the theory of modules over cm-monoids, offering a unified approach to polymorphisms and invariant relations, in the hope of providing new insights into algebraic structures and CSP complexity theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
