Hyperidentities in Graph Variety Generated by Zeropotent and Unipotent Graphs

Abstract

Directed graphs without multiple edges can be represented as algebras of type (2,0), so-called graph algebras. We say that a graph 𝐺 satisfies a term equation 𝑠≈𝑡 if the corresponding graph algebra 𝐴(𝐺) satisfies 𝑠≈𝑡. The set of all term equations 𝑠≈𝑡 which the graph 𝐺 satisfies denoted by Id({𝐺}). The class of all graph algebras satisfying all term equations in Id({𝐺}) is called the graph variety generated by 𝐺, denoted by 𝒱𝑔({𝐺}). A term equation 𝑠≈𝑡 is called an identity in 𝒱𝑔({𝐺}) if 𝐴(𝐺) satisfies 𝑠≈𝑡 for all 𝐺 ∈ 𝒱𝑔({𝐺}). An identity 𝑠≈𝑡 of terms 𝑠𝑠 and 𝑡 of any type 𝜏 is called a hyperidentity of an algebra if whenever the operation symbols occurring in 𝑠𝑠 and 𝑡𝑡 are replaced by any term operations of of the appropriate arity, the resulting identities hold in . In this paper, we characterize all identities, all graphs and all hyperidentities in 𝒱𝑔({𝐺}) where 𝐺 is the zeropotent and unipotent.