Numbers implying algebraic structures

We prove in this paper that a commutative idempotent groupoid is a proper P lonka 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 2 7 is the characteristic number of those sums in the variety of all commutat ive idempotent groupoids. Using this result, we also show that a medial ide mpotent groupoid with three essentially binary operations contains a subgro upoid 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).