Representations of Menger hypercompositional algebras by some types of commutative hyperoperations

Abstract

We present an abstract characterization of diagonal semihypergroups derived from any Menger hypercompositional algebra. We also prove that the set of all $k$-commutative hyperoperations forms a Menger algebra. The necessary and sufficient conditions under which a Menger hypercompositional algebra of rank $n>1$ is embeddable into an algebra of $k$-commutative hyperoperations are proposed.