多态同质性与泛代数几何
Polymorphism-homogeneity and universal algebraic geometry
AI总结：
本文利用方程解集为有限代数赋予关系结构，证明多态同质性等价于代数集为中心化子克隆下封闭的元组集，且等价于代数在其有限次幂范畴中内射，并刻画了满足条件的有限半格、格、阿贝尔群和单酉代数。
摘要：
我们以规范的方式，利用方程的解集为任意有限代数赋予一个关系结构，并证明该关系结构是多态同质的当且仅当该代数本身是多态同质的。我们表明，多态同质性也等价于以下性质：代数集（即方程组的解集）恰好是那些在代数的中心化子克隆下封闭的元组集合。此外，我们证明上述性质成立当且仅当该代数在其有限次幂范畴中是内射的。我们还考虑了两个附加条件：多态同质性和内射性的更强变体，并明确刻画了满足这三个条件中任意一个的有限半格、格、阿贝尔群和单酉代数。
英文摘要：
We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions.
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。
