Numbers implying algebraic structures

Abstract

We prove in this paper that a commutative idempotent groupoid is a proper Plonka sum of affine spaces over GF(3) if and only if it has 27 essentially 4-ary term functions. This means, in another terminology, that the number 27 is the characteristic number of those sums in the variety of all commutative idempotent groupoids. Using this result, we also show that a medial idempotent groupoid with three essentially binary operations contains a subgroupoid term equivalent to either the five-element affine space over GF(5) or a Plonka sum of a trivial groupoid and the three-element affine space over GF(3).