Classification of minimal clones over a finite field in multiple-valued logic

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For a fixed set A, a set of multi-variable functions defined over A is called a clone if it contains all projections and is closed with respect to (functional) composition. The set of all clones on A has the structure of a lattice. An atom of the lattice of clones is called a minimal clone. In this research, we introduced the structure of a finite field into the base set A and considered generating functions of minimal clones as polynomials over the finite field A. Under this new point of view, we aimed at classifying minimal clones.For minimal clones which are generated by binary idempotent functions or ternary majority functions, we have succeeded to find the polynomial expressions of their minimal functions for arbitrary A with the cardinality of prime power.