On n-groupoids defined on fields

Abstract

Let us define n-ary operations f on the real field R and the Galois fields GF(p) by f(x ₁, x ₂, ···, x _n)=α ₁ x ₁+α ₂ x ₂+···+α _n x _n with α ₁, α ₂, ···, α _n, none of them zero, in the fields. Then we obtain n-groupoids. Let B be the class of all n-groupoids defined on Galois fields in this way. In this paper, we will show that the equational theory of B is precisely that of (R,f) if and only if α ₁, α ₂, ···, α _n, as elements of R, are algebraically independent. If we restrict our attention only to the case when all α _i's are equal to α, then the equational theory of B is exactly that of (R,f) if and only if α is transcendental as an element of R. Unlike the groupoid cases ([1], [5]) and the case when α ₁+α ₂+···+α _n=1 ([2]), the basis problems for the equational theories of the above n-groupoids or the above classes are not so simple.