$S(\bar{n_i},Y_{i})$-Terms and Their Algebraic Properties

Sarawut Phuapong, Chollawat Pookpienlert

Abstract

A clone is a set of operations defined on a base set which is closed under composition and contains all projection operations. A  special kind of clone satisfies the superassociative law is called a Menger algebra. In this paper, we introduce the new concept of $S(\bar{n_i},Y_{i})$-terms of type $\tau$. The set of all $S(\bar{n_i},Y_{i})$-terms of type $\tau$ is closed under the superposition operation $S^{n_{i}}$ and so forms a clone denoted by $clone_{S(\bar{n_i},Y_{i})}(\tau)$. We show that the $clone_{S(\bar{n_i},Y_{i})}(\tau)$ is a Menger algebra of rank $n_{i}$and study its algebraic properties. A connection between identities in $clone_{S(\bar{n_i},Y_{i})}(\tau)$ and $S(\bar{n_i},Y_{i})$-hyperidentities is established.