Linear-Hypersubstitutions for Algebraic Systems of Type ((n); (n)) and Characterization of Their Idempotent Elements

Jintana Joomwong, Dara Phusanga

Abstract

A formula in which each variable occurs at most once is said to be a linear-formula ([1, 2]). A linear-hypersubstitution for algebraic systems of type $((n);(n))$ is a mapping $\sigma_{_{\small{t,F}}}$ which maps $n$-ary operation symbols $f$ to $n$-ary linear-terms $\sigma_{_{\small{t,F}}}(f)$ and $n$-ary relational symbols $\gamma$ to $n$-ary linear-formulas $\sigma_{_{\small{t,F}}}(\gamma)$. Any linear-hypersubstitution $\sigma_{\small{t,F}}$ can be extended to a mapping $\widehat{\sigma}_{_{\small{t,F}}}$ on the set of all linear- terms of type $(n)$ and linear-formulas of type $((n);(n))$. A binary operation “$\circ_{_{\small{lin}}}$” on $Hyp^{lin}((n);(n))$ the set of all linear-hypersubstitutions for algebraic systems of type $((n);(n))$ can be defined by using this extension. The set $Hyp^{lin}((n);(n))$ together with the identity linear-hypersubstitution $(\sigma_{_{\small{t,F}}})_{_{\small{id}}}$ which maps $(\sigma_{_{\small{t,F}}})_{_{id}}(f):=f(x_{1},\ldots,x_{n})$ and $(\sigma_{_{t,F}})_{_{\small{id}}}(\gamma):=\gamma(x_{1},\ldots,x_{n})$ forms a monoid. The concept of an idempotent element plays an important role in semigroup theory [3]. In this paper, we characterize the idempotent of the monoid of linear-hypersubstitutions for algebraic systems of type $((n);(n))$.