HYPERIDENTITIES IN (xy)x ≈x(yy) GRAPH ALGEBRAS OF TYPE (2,0)

Abstract

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