Identities in (x(yz))z with Opposite Loop and Reverse Arc Graph Varieties of Type (2,0)

Amporn Anantpinitwatna, Tiang Poomsa-ard

Abstract

Graph algebrasestablish a connection between directed graphs without multipleedges and special universal algebras of type (2,0). We say that agraph $G$ satisfies a term equation $s \approx t$ if thecorresponding graph algebra $\underline{A(G)}$ satisfies $s \approxt$. A class of graph algebras $\mathcal{V}$ is called a graphvariety if $\mathcal{V} = Mod \Sigma$ where $\Sigma$ is a subset of$T(X) \times T(X)$. A term equation $s \approx t$ is called anidentity in a graph variety $V$ if $G$ satisfies $s \approx t$ forall $G\in \mathcal{V}$. A graph variety $\mathcal{V'} =Mod\Sigma^{'}$ is called an $(x(yz))z$ with opposite loop andreverse arc graph variety if $\Sigma^{'}$ is a set of $(x(yz))z$with opposite loop and reverse arc term equations. In this paper wecharacterize identities in each (x(yz))z with opposite loop andreverse arc graph variety.