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

对于固定集合A，定义在A上的多变量函数的集合称为克隆，如果它包含所有投影并且相对于(泛函)合成是闭的。A上所有克隆的集合具有格的结构。克隆格子中的一个原子称为极小克隆。将有限域的结构引入到基集A中，将极小克隆的生成函数看作有限域A上的多项式，在这种新的观点下，我们对极小克隆进行了分类，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.