Algebraic properties of Menger Hyperalgebras and applications

Abstract

For a fixed positive n, an algebra of type (n + 1) is called a Menger algebra if the (n + 1)-ary operation defined on the base set of algebra satisfies the superasscociativity. The core idea of this thesis is the construction of a generalization of Menger algebras, especially further development in the theory of Menger algebras and semihypergroups along with some applications to theoretical computer science. The results of this thesis is divided into three main parts. In the first part, we focus on Menger algebras. Fundamental constructions and several properties of Menger algebras are given. Specific classes of n-ary functions are introduced, and a characterization for any (n + 1)-ary groupoids to be Menger algebras is proved. Generally, semihypergroups are algebraic hyperstructures that generalized semigroups. For this reason, in the second part of this thesis, we introduce the novel concept of Menger hyperalgebras, which is a natural generalization of both semihypergroups and Menger algebras. Several descriptions of certain algebraic properties including subhyperalgebras, homomorphisms, and quotient hyperstructures, are provided. Based on representation theorems for semigroups and Menger algebras, the situation in both semihypergroups and Menger hyperalgebras are studied using the idea of multivalued transformations and their extensions. Finally, in view of applications, we construct the Menger algebra of terms rewriting systems consisting of the set of all rewrite rules and an (n + 1)-ary operation satisfying the superassociativity by applying the theory which is developed in the first two parts of this thesis. A nice connection between terms that appear in the study of universal algebra and terms rewriting systems is also described.