Studies on the structure of the lattice of clones consisting of multiple-valued logical functions

克隆人是一组K值逻辑函数，它们被封闭在组成下，并包含所有的预测。所有clones consisting of k-v\ed logical functions is denoted by L-D2k-D2。Whereas the structure of LイイD22イエD2 is·etely determined， our knowledge about the structure of LイイD2kイエD2 for k > 2 at present is very little。本研究的主要目标是澄清L-D2 k-D2的结构，我们获得了以下结果。The structure of LイD23イイD2 as a metric spaceWe have introduced a metric into the lattice LイD2kイD2 of clones and showed that LイD2kイD2 is a compact metric space。Moreover， we constructed continuous mappings， based on the meet operation， from LイイD23イエD2 onto LイD22イエD2 and studied the images of max·clones in LイD23イエD2 and those of some clones being accumulation points under such mappings. 2。L-D2 K-D2及相关主题中的最低限度克隆人自从最低限度克隆人分类以来还远远没有完成，研究的结果 ... More 最小克隆人的各种属性是非常重要的。We have studed a particular proManagement concerning min？clones： Given a pair （C-D21-D2，C-D22-D2） of min？clones， we call it gigantic pair if the union C-D21-D22-D2 generates the whole set of functions。我们提出了一个关于巨人对的角色化理论，并展示了大多数K's存在的巨型对。最近的超克隆研究， I。G..罗森伯格发起了超克隆人的研究。我们继续他的工作，并提供了后续工作：集合{0，1}上所有超克隆人的晶格特征是连续的。This is interesting as the cardinality of the lattice of all （ordinary） clones on {0， 1} is countable.4。对部分克隆人与部分功能一致性的研究，我们调查了部分克隆人的后续问题： （1）最大部分克隆人的最小数目，谁相遇就是次要部分克隆人。（2）最小部分克隆人的最小数目-它们的联接是所有部分操作的克隆人。这是与L教授的联合工作。Haddad and I.G.罗森伯格。Less（低）

A clone is a set of k-valued logical functions which is closed under composition and contains all the projections. The set of all clones consisting of k-valued logical functions is denoted by LィイD2kィエD2. Whereas the structure of LィイD22ィエD2 is completely determined, our knowledge about the structure of LィイD2kィエD2 for k > 2 at present is very little. The main objective of this research is to clarify the structure of LィイD2kィエD2 and we have obtained the following results.1. The structure of LィイD23ィエD2 as a metric spaceWe have introduced a metric into the lattice LィイD2kィエD2 of clones and showed that LィイD2kィエD2 is a compact metric space. Moreover, we constructed continuous mappings, based on the meet operation, from LィイD23ィエD2 onto LィイD22ィエD2 and studied the images of maximal clones in LィイD23ィエD2 and those of some clones being accumulation points under such mappings.2. Minimal clones in LィイD2kィエD2 and related topicsSince the classification of minimal clones is far from complete, the study of … More various properties of minimal clones are very important. We have studied a particular problem concerning minimal clones: Given a pair (CィイD21ィエD2, CィイD22ィエD2) of minimal clones, we call it gigantic pair if the union CィイD21ィエD2∪CィイD22ィエD2 generates the whole set of functions. We proved a characterization theorem of gigantic pairs and showed that gigantic pairs exist for most k's.3. Study of hyperclonesRecently, I. G. Rosenberg initiated the study of hyperclones. We continued his work and proved the following: The cardinality of the lattice of all hyperclones on the set {0,1} is of continuum. This is interesting as the cardinality of the lattice of all (ordinary) clones on {0, 1} is countable.4. Study of partial clones consisting of partial functionsWe investigated the following problems on partial clones : (1) The minimal number of maximal partial clones whose meet is the trivial partial clone. (2) The minimal number of minimal partial clones whose join is the clone of all partial operations. This is a joint work with Professors L. Haddad and I.G. Rosenberg. Less