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

对于固定集合a，定义在a上的多变量函数集，如果它包含所有的投影，并且对（泛函）复合是封闭的，则称为克隆。A上所有克隆的集合具有晶格结构。克隆晶格中的原子称为最小克隆。本文将有限域的结构引入基集a中，并将最小克隆的生成函数视为有限域a上的多项式。在此观点下，我们旨在对最小克隆进行分类。对于由二元幂等函数或三元多数函数生成的极小克隆，我们成功地找到了它们的极小函数在任意A下的素数幂的多项式表达式。

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.