Monoid of Cohypersubstitutions of Type $\tau=(n)$

Darunee Boonchari, Kittisak Saengsura

Abstract

A mapping $\sigma$ which assigns to every $n_i$-ary cooperation symbol $f_i$ an $n_i$-ary coterm of type $\tau=(n_i)_{i\in I}$ is said to be a cohypersubstitution of type $\tau$. The concepts of cohypersubstitutions were introduced by K.Denecke in [4]. Every cohypersubstitution $\sigma$ of type $\tau$ induces a mapping $\hat\sigma$ on the set of all coterms of type $\tau$. The set of all cohypersubstitutions of type $\tau$ under the binary operation $\hat\circ$ which is defined by $\sigma_1\hat\circ\sigma_2:=\hat\sigma_1\circ\sigma_2$ for all $\sigma_1,\sigma_2\in Cohyp(\tau)$ forms a monoid which is called the monoid of cohypersubstitution of type $\tau$. In this research, we characterize all idempotent and regular elements of $Cohyp(n)$ and characterize some Green's relations $L$ and $R$ on $Cohyp(n)$.