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

• Volume: 29, Issue: 2, page 81-107
• ISSN: 1509-9415

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## Abstract

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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 $\underline{A\left(G\right)}$ satisfies s ≈ t. A class of graph algebras V is called a graph variety if $V=Mo{d}_{g}\Sigma$ where Σ is a subset of T(X) × T(X). A graph variety ${V}^{\text{'}}=Mo{d}_{g}{\Sigma }^{\text{'}}$ 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 $\underline{A\left(G\right)}$ satisfies s ≈ t for all G ∈ V. An identity s ≈ t of a variety V is called a hyperidentity of a graph algebra $\underline{A\left(G\right)}$, G ∈ V whenever the operation symbols occuring in s and t are replaced by any term operations of $\underline{A\left(G\right)}$ of the appropriate arity, the resulting identities hold in $\underline{A\left(G\right)}$. An identity s ≈ t of a variety V is called an M-hyperidentity of a graph algebra $\underline{A\left(G\right)}$, 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 $\underline{A\left(G\right)}$ of the appropriate arity, the resulting identities hold in $\underline{A\left(G\right)}$. 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].

## How to cite

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Apinant Anantpinitwatna, and Tiang Poomsa-ard. "Special m-hyperidentities in biregular leftmost graph varieties of type (2,0)." Discussiones Mathematicae - General Algebra and Applications 29.2 (2009): 81-107. <http://eudml.org/doc/276824>.

@article{ApinantAnantpinitwatna2009,
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 a term equation s ≈ t if the corresponding graph algebra $\underline\{A(G)\}$ satisfies s ≈ t. A class of graph algebras V is called a graph variety if $V = Mod_g Σ$ where Σ is a subset of T(X) × T(X). A graph variety $V^\{\prime \} = Mod_gΣ^\{\prime \}$ 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 $\underline\{A(G)\}$ satisfies s ≈ t for all G ∈ V. An identity s ≈ t of a variety V is called a hyperidentity of a graph algebra $\underline\{A(G)\}$, G ∈ V whenever the operation symbols occuring in s and t are replaced by any term operations of $\underline\{A(G)\}$ of the appropriate arity, the resulting identities hold in $\underline\{A(G)\}$. An identity s ≈ t of a variety V is called an M-hyperidentity of a graph algebra $\underline\{A(G)\}$, 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 $\underline\{A(G)\}$ of the appropriate arity, the resulting identities hold in $\underline\{A(G)\}$. 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].},
author = {Apinant Anantpinitwatna, Tiang Poomsa-ard},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {varieties; biregular leftmost graph varieties; identities; term; hyperidentity; M-hyperidentity; binary algebra; graph algebra; -hyperidentity},
language = {eng},
number = {2},
pages = {81-107},
title = {Special m-hyperidentities in biregular leftmost graph varieties of type (2,0)},
url = {http://eudml.org/doc/276824},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Apinant Anantpinitwatna
AU - Tiang Poomsa-ard
TI - Special m-hyperidentities in biregular leftmost graph varieties of type (2,0)
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2009
VL - 29
IS - 2
SP - 81
EP - 107
AB - 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 $\underline{A(G)}$ satisfies s ≈ t. A class of graph algebras V is called a graph variety if $V = Mod_g Σ$ where Σ is a subset of T(X) × T(X). A graph variety $V^{\prime } = Mod_gΣ^{\prime }$ 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 $\underline{A(G)}$ satisfies s ≈ t for all G ∈ V. An identity s ≈ t of a variety V is called a hyperidentity of a graph algebra $\underline{A(G)}$, G ∈ V whenever the operation symbols occuring in s and t are replaced by any term operations of $\underline{A(G)}$ of the appropriate arity, the resulting identities hold in $\underline{A(G)}$. An identity s ≈ t of a variety V is called an M-hyperidentity of a graph algebra $\underline{A(G)}$, 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 $\underline{A(G)}$ of the appropriate arity, the resulting identities hold in $\underline{A(G)}$. 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].
LA - eng
KW - varieties; biregular leftmost graph varieties; identities; term; hyperidentity; M-hyperidentity; binary algebra; graph algebra; -hyperidentity
UR - http://eudml.org/doc/276824
ER -

## References

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