Idempotent elements of $WP_G(2,2)\cup\{\sigma_{id}$

Sorasak Leeratanavalee

Abstract. A generalized hypersubstitution of type $\tau=(2,2)$ is a mapping $\sigma$ which maps the binary operation symbols $f$ and $g$ to terms $\sigma(f)$ and $\sigma(g)$ which does not necessarily preserve arities. Any generalized hypersubstitution $\sigma$ can be extended to a mapping $\hat{\sigma}$ on the set of all terms of type $\tau=(2,2)$. A binary operation on $Hyp_G(2,2)$ the set of all generalized hypersubstitutions of type $\tau=(2,2)$ can be defined by using this extension. The set $Hyp_G(2,2)$ together with the identity hypersubstitution $\sigma_{id}$ which maps $f$ to $f(x_1, x_2)$ and maps $g$ to $g(x_1, x_2)$ forms a monoid. The concept of an idempotent element plays an important role in many branches of mathematics, for instance, in semigroup theory and semiring theory. In this paper we characterize the idempotent generalized hypersubstitutions of $WP_G(2,2)\cup\{\sigma_{id}\}$ a submonoid of $\underline{Hyp_G(2,2)}$. SAMO NASLOV