Hyperidentities in $(xy)x approx x(yy)$ graph algebras of type $(2,0)$

Bull. Korean Math. Soc. 2007 Vol. 44, No. 4, 651-661 Published online December 1, 2007

Jeeranunt Khampakdee and Tiang Poomsa-ard Mahasarakham University, Mahasarakham University

Abstract : Graph algebras establish a connection between directed gra-phs without multiple edges and special universal algebras of type $(2,0)$. We say that a graph $G$ satisfies an identity $s \approx t$ if the corresponding graph algebra $\underline{A(G)}$ satisfies $s \approx t$. A graph $G = (V,E)$ is called an $(xy)x \approx x(yy)$ graph if the graph algebra $\underline{A(G)}$ satisfies the equation $(xy)x \approx x(yy)$. An identity $s \approx t$ of terms $s$ and $t$ of any type $\tau$ is called a hyperidentity of an algebra $\underline {A}$ if whenever the operation symbols occurring in $s$ and $t$ are replaced by any term operations of $\underline{A}$ of the appropriate arity, the resulting identities hold in $\underline {A}$. In this paper we characterize $(xy)x\approx x(yy)$ graph algebras, identities and hyperidentities in $(xy)x\approx x(yy)$ graph algebras.

Keywords : identities, hyperidentities, term, normal form term, binary algebra, graph algebra, $(xy)x\approx x(yy)$ graph algebra