An Approach Toward Classification of Minimal Groupoids on a Finite Set

A minimal groupoid is a minimal clone generated by a binary idempotent function. The classification of minimal groupoids on a finite set is not yet complete and seems to be quite a hard task. In this paper a new viewpoint is proposed toward the classification of minimal groupoids. The pr-distance is introduced for binary functions. Using this concept, the generators of 48 minimal groupoids on a 3-element set are classified into three classes: Commutative functions, functions with pr-distance 1 and those with pr-distance 2. Some of the results obtained for the 3-element case generalize to any finite case. In particular, a binary idempotent function on any finite set is proved to generate a minimal groupoid if its pr-distance is 1.