Hyperidentities in Graph Variety Generated by Zeropotent and Unipotent Graphs

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Chaowat Manyuen


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 gif.latex?\underline{A} if whenever the operation symbols occurring in 𝑠𝑠 and 𝑡𝑡 are replaced by any term operations of gif.latex?\underline{A} of the appropriate arity, the resulting identities hold in gif.latex?\underline{A}. In this paper, we characterize all identities, all graphs and all hyperidentities in 𝒱𝑔({𝐺}) where 𝐺 is the zeropotent and unipotent.

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Manyuen, C. (2023). Hyperidentities in Graph Variety Generated by Zeropotent and Unipotent Graphs. Mathematical Journal by The Mathematical Association of Thailand Under The Patronage of His Majesty The King, 68(709), 22–36. Retrieved from https://ph02.tci-thaijo.org/index.php/MJMATh/article/view/244566
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