Classification of Dual Distributive Hyperidentities in Divisible Algebras

Abstract

The paper provides a classification of nontrivial dual hyperidentities of the left and right distributivity satisfied in functionally nontrivial divisible algebras. If the nontrivial dual hyperidentities of the left and right distributivity hold in a functionally nontrivial divisible algebra, then the hyperidentity of the left distributivity is of rank two and is (equivalent to the hyperidentity) of the form

$$X(x,Y(y,z))=Y(X(x,y),X(x,z)),$$

while the hyperidentity of the right distributivity is the hyperidentity of rank two and is (equivalent to the hyperidentity) of the form

$$X(Y(x,y),z)=Y(X(x,z),X(y,z)).$$

For the classification of nontrivial hyperidentities of the left and right distributivity satisfying in functionally nontrivial \(q\)-algebras, see [1–4].