Outermost-Strongly Solid Variety of Commutative Semigroups

Identities are used to classify algebras into collections called varieties, hyperidentities are used to classify varieties into collections called hypervarieties. Hyperidentities have an interpretation in the theory of switching circuits and are also closely related to clone theory. The tool used to study hyperidentities is the concept of a hypersubstitution, see [1]. The generalized concept of a hypersubstitution is a generalized hypersubstitution. Generalized hypersubstitutions are mappings from the set of all fundamental operations into the set of all terms of the same language, which need not necessarily preserve the arities. Identities which are closed under generalized hypersubstitutions are called strong hyperidentities. A variety in which each of its identity is a strong hyperidentity is called strongly solid. In this paper we study a submonoid of the monoid of all generalized hypersubstitutions which is called the monoid of all outermost generalized hypersubstitutions and determine the greatest outermost-strongly solid variety of commutative semigroups.