We investigate finitary functions from ℤn to ℤn for a square-free number n. We show that the lattice of all clones on the square-free set ℤp1⋯pm which contain the addition of ℤp1⋯pm 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 (ℤpi, 𝔽i)-linearly closed clonoids, ℒ(ℤpi, 𝔽i), to the pi + 1 power, where 𝔽i =∏j∈{1,…,m}∖{i}ℤpj. These lattices are studied in [S. Fioravanti, Closed sets of finitary functions between products of finite fields of pair-wise coprime order, preprint (2020), arXiv:2009.02237] and there we can find an upper bound for their cardinality. Furthermore, we prove that these clones can be generated by a set of functions of arity at most max(p1,…,pm).

中文翻译：

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阿贝尔无平方群的展开

我们从ℤn到ℤn对于一个无平方数n. 我们证明了无平方集上所有克隆的格ℤp1⋯p米其中包含添加ℤp1⋯p米是有限的。我们通过对所有格的直接乘积的单射函数为这个格的基数提供一个上限(ℤp一世, 𝔽一世)-线性闭合的克隆体，ℒ(ℤp一世, 𝔽一世), 对p一世 + 1电源，在哪里𝔽一世 =∏j∈{1,…,米}∖{一世}ℤpj. 这些晶格在 [S. Fioravanti，成对互质序的有限域乘积之间的封闭有限函数集，预印本 (2020)，arXiv:2009.02237]，我们可以在那里找到它们基数的上限。此外，我们证明这些克隆最多可以由一组arity函数生成最大限度(p1,…,p米).