阿贝尔无平方因子群的扩张
Expansions of abelian squarefree groups
AI总结：
该研究针对无平方因子数对应的模n整数集，证明了含加法运算的克隆格有限，给出其基数上界的构造方式，并证明这类克隆可由元数不超过各素因子最大值的函数集生成，推进了阿贝尔无平方因子群相关克隆结构的研究。
摘要：
我们研究了无平方因子数n下从ℤₙ到ℤₙ的有限性函数。我们证明了，无平方因子集合ℤ_{p₁⋯pₘ}上所有包含ℤ_{p₁⋯pₘ}加法运算的克隆构成的格是有限的。我们通过一个单射给出了该格基数的上界：该单射将其映射到所有(ℤ_{pᵢ}, 𝔽ᵢ)-线性闭克隆oid的格ℒ(ℤ_{pᵢ}, 𝔽ᵢ)的直积的pᵢ+1次幂，其中𝔽ᵢ = ∏_{j ∈ {1,…,m}\{i}} ℤ_{pⱼ}。这些格已有相关研究，我们可以得到其基数的上界。此外，我们证明这些克隆可由元数至多为max(p₁,…,pₘ)的函数集生成。
英文摘要：
We investigate finitary functions from $\mathbb{Z}_{n}$ to $\mathbb{Z}_{n}$ for a squarefree number $n$. We show that the lattice of all clones on the squarefree set $\mathbb{Z}_{p_1\cdots p_m}$ which contain the addition of $\mathbb{Z}_{p_1\cdots p_m}$ is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattices of all $(\mathbb{Z}_{p_i}, \mathbb{F}_i)$-linearly closed clonoids, $\mathcal{L}(\mathbb{Z}_{p_i}, \mathbb{F}_i)$, to the $p_i+1$ power, where $\mathbb{F}_i = \prod_{j \in \{1,\dots,m\}\backslash \{i\}}\mathbb{Z}_{p_j}$. These lattices are studied in the litterature and we can find an upper bound for cardinality of them. Furthermore, we prove that these clones can be generated by a set of functions of arity at most $\max(p_1,\dots,p_m)$.
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。
