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

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 I D2k D2.Whereas the structure of L-I D22-D2is completely determined，our knowledge about the structure of L-I D2k-D2for k&gt；2at present is very little。The main objective of this research is to clarify the structure of L I D2k锡D2 and we have obtained the following results.1。The structure of L I D 23 I D 2 as a metric spaceWe have introduced a metric into the lattice L I D 2 k D 2 of clones and showed that L I D 2 k D 2 is a compact metric space.Moreover，we constructed continuous mappings，based on the meet operation，from L I D23文件D2 onto L文件D22文件D2 and studied the images of maximal clones in L文件D23文件D2 and those of some clones being accumulation points under such mappings.2。Minimal clones in L I D2k ii 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 I D 21 I D 2，C I D 22 D 2)of minimal clones，we call it gigantic pair if the union C I D 21 I D 2áC I D 22 D 2 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 clones。(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：Less

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