Algebraic connections between Menger algebras and Menger hyperalgebras via regularity

Abstract

Menger hyperalgebras of rank \(n\), where \(n\) is a fixed integer, can be regarded as a natural generalization of arbitrary semihypergroups. Based on this knowledge, an interesting question arises: what a generalization of regular semihypergroups is. In the article, we establish the notion of \(v\)-regular Menger hyperalgebras of rank \(n\), which can be considered as an extension of regular semihypergroups. Furthermore, we study regularity of Menger hyperalgebras of rank \(n\) which are induced by some subsets of Menger algebras of rank \(n\). In particular, we obtain sufficient conditions so that the Menger hyperalgebras of rank \(n\) are \(v\)-regular.