Special m-hyperidentities in biregular leftmost graph varieties of type (2,0)

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 a term equation s ≈ t if the corresponding graph algebra ${\displaystyle \underline{A\left(G\right)}}$ satisfies s ≈ t. A class of graph algebras V is called a graph variety if ${\displaystyle V=Mo{d}_g\Sigma }$ where Σ is a subset of T(X) × T(X). A graph variety ${\displaystyle V\prime =Mo{d}_g\Sigma \prime }$ is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if ${\displaystyle \underline{A\left(G\right)}}$ satisfies s ≈ t for all G ∈ V. An identity s ≈ t of a variety V is called a hyperidentity of a graph algebra ${\displaystyle \underline{A\left(G\right)}}$, G ∈ V whenever the operation symbols occuring in s and t are replaced by any term operations of ${\displaystyle \underline{A\left(G\right)}}$ of the appropriate arity, the resulting identities hold in ${\displaystyle \underline{A\left(G\right)}}$. An identity s ≈ t of a variety V is called an M-hyperidentity of a graph algebra ${\displaystyle \underline{A\left(G\right)}}$, G ∈ V whenever the operation symbols occuring in s and t are replaced by any term operations in a subgroupoid M of term operations of ${\displaystyle \underline{A\left(G\right)}}$ of the appropriate arity, the resulting identities hold in ${\displaystyle \underline{A\left(G\right)}}$. In this paper we characterize special M-hyperidentities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [3].