On idempotent and hyperassociative structures (Q2010165)

An algebra with binary operations is called a binary algebra. A binary algebra \((Q;\Sigma)\) is said to be 1) hyperassociative if \(X(x,Y(y,z))=Y(X(x,y),z)\) for every operations \(X,Y\in\Sigma\), 2) rectangular if \(X(x,X(y,x))=x\) for every operation \(X\in\Sigma\). The main result of the paper under review is Theorem 4. Let \((Q;\Sigma)\) be an idempotent and hyperassociative binary algebra. Then if there exists on it such a congruence that corresponding quotient algebra is a rectangular semigroup and operations of the set are semilattice operations on each equivalence class then \((Q;\Sigma)\) satisfies the transitive commutativity property \(X(x,y)=X(y,x)\) and \(X(y,z)=X(z,y)\Rightarrow X(x,z)=X(z,x)\).