Abstract

The main problem of clone theory is to describe the clone lattice for a given basic set. For a two-element basic set this was resolved by E.L. Post, but for at least three-element basic set the full structure of the lattice is still unknown, and the complete description in general is considered to be hopeless. Therefore, it is studied by its substructures and its approximations. One of the possible directions is to examine k-ary parts of the clones and their mutual inclusions. In this paper we study k-ary parts of maximal clones, for \(k\geqslant 2,\) building on the already known results for their unary parts. It turns out that the poset of k-ary parts of maximal clones defined by central relations contains long chains.

中文翻译：

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关于最大克隆的k元部分

摘要

克隆理论的主要问题是描述给定基本集的克隆格。对于二元素基本集，EL Post 解决了这个问题，但对于至少三元素基本集，格子的完整结构仍然未知，并且一般认为完整的描述是没有希望的。因此，通过其子结构及其近似来研究它。可能的方向之一是检查克隆的k部分及其相互包含物。在本文中，我们研究最大克隆的k元部分，对于\(k\geqslant 2,\)建立在其一元部分的已知结果的基础上。事实证明，由中心关系定义的最大克隆的k元部分的偏序集包含长链。