A Study of Generalized Clones of Rank $k$ and Generalized $k$-Hypersubstitutions

Nareupanat Lekkoksung, Prakit Jampachon

Abstract

The set of all $n$-ary terms of type $\tau_{n}$ together with an $(n+1)$-ary superposition and $n$ nullary operation symbols forms an algebra, so-called a unitary Menger algebra of rank $n$.Generalizing this idea, let $k \geq n$, we study an algebraic structure consisting of the set of all $k$-ary terms of type $\tau_{n}$, an $(n+1)$-ary generalized superposition and $k$ nullary operation symbols.We call this algebra a generalized clone of rank $k$.We show that the generalized clone of rank $k$ is a unitary Menger algebra of rank $k$.We use this concept to investigate the properties of a particular generalized hypersubstitution of type $\tau_{n}$ which maps each operation symbol of type $\tau_{n}$ to a $k$-ary term of the same type.