Theory of commutation and minimal clones in multiple-valued logic

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A set of multi-variable functions defined on a given set A is a clone on A if it is closed under functional composition and contains all projections on A. The set of clones on A forms a lattice. Except the case for |A|=2, the structure of the clone lattice on A is extremely complex and, until now, mostly unknown.Based on the commutativity of functions the notion of a centralizer (centralizing clone) is defined and, furthermore, a centralizing monoid is defined as the set of unary functions of a centralizer. In this project, for the case of |A|=3, we determined all centralizing monoids on A as well as the inclusion relations among them. We also investigated the relation between maximal centralizing monoids and majority functions which generate minimal clones.