Clones of many-valued logic functions (esp.semirigidity problems)

1.给出了基于n-立方体下半部相交反链个数的团(或交)布尔函数个数的显式公式。通过对参数的计算机估计，我们得到了七个变量的团函数的个数。(参考文献1)2.我们证明了n-立方体的下半部分的相交反链集合与(n-1)-立方体的相交反链集合之间存在1-1对应。这将包含在前者集合中的相交反链的计数减少到后者中。(参考文献2)3.半刚性集的研究起源于基的分类。在这个仅对*A*=2，3完全解决的复杂问题中，任务之一是寻找极大克隆系统的所有极小非平凡交。大多数克隆是由自反关系(二元或更高的)确定的，因此我们需要确定这些关系的子集R，使得每个保持R中所有关系的函数要么是常数，要么是投影。在这篇文章中，我们简要地概述了1)保序关系，2)中心关系和3)拟线性关系。最后，我们补充了4)自对偶克隆的一些新结果，证明了对于k素数，每两个极大自对偶克隆的基是刚性的，即它们只共享单位映射。(参考文献3)

1. We gave an explicit formula for the number of clique (or intersecting) Boolean functions in terms of the parameters based upon the number of intersecting antichains in the lower half of the n-cube. We obtained the numbers of clique functions up to seven variables through computer evaluations of the parameter. (Reference 1)2. We proved that there is a 1-1 correspondence between the set of intersecting antichains in the lower-half of the n-cube and the set of intersecting antichains in the (n-1)-cube. This reduces the enumeration of intersecting antichains contained in the former set to that in the latter. (Reference 2)3. The study of semirigid sets arose from the classification of bases. In this complex problem-fully solved only for *A*=2, 3-one of the tasks is to find all minimal nontrivial intersections of systems of maximal clones. Most of the clones are determined by reflexive relations (binary or of higher arities) and so we need to determine subsets R of these relations such that every function preserving all relations in R is either constant or is a projection. In this paper we give a short overview of this problem for 1) isotone relations, 2) central relations and 3)quasi-linear relations. Finally we add some new results for 4) autodual clones ; we proved that for k prime the foundations of every two maximal autodual clones are rigid, i.e. they share only the identity map. (Reference 3)