Characterization of Some Regular Relational Hypersubstitutions for Algebraic Systems

Abstract

An algebraic system is a structure which consists of a nonempty set together with a sequence of operations and a sequence of relations which are defined on the set. The concept of a relational hypersubstitution for algebraic systems is a canonical extension of the concept of a hypersubstitution for universal algebras. Such relational hypersubstitutions are mappings which map operation symbols to terms and map relation symbols to relational terms preserving arities. The set of all relational hypersubstitutions for algebraic systems together with an associative binary operation, which was defined in [D. Phusanga, J. Koppitz, The monoid of hypersubstitutions for algebraic systems, Announcements of Union of Scientists Silven 33 (1) (2018) 119-126], forms a monoid. The concept of the special regular elements are important role in semigroup theory. In this paper, we characterize the set of all completely regular, left regular and right regular elements of this monoid of type $((m),(n))$. The results show that the set of all completely regular elements and the set of all left(right) regular elements of this monoid are the same.