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

Apinant Anantpinitwatna; Tiang Poomsa-ard

Discussiones Mathematicae - General Algebra and Applications (2009)

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

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 A ( G ) ̲ satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = M o d g Σ where Σ is a subset of T(X) × T(X). A graph variety V ' = M o d g Σ ' 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 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 A ( G ) ̲ , G ∈ V whenever the operation symbols occuring in s and t are replaced by any term operations of A ( G ) ̲ of the appropriate arity, the resulting identities hold in A ( G ) ̲ . An identity s ≈ t of a variety V is called an M-hyperidentity of a graph algebra 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 A ( G ) ̲ of the appropriate arity, the resulting identities hold in 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].

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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  1. [1] A. Ananpinitwatna and T. Poomsa-ard, Identities in biregular leftmost graph varieties of type (2,0), Asian-European J. of Math. 2 (1) (2009), 1-18. Zbl1201.08006
  2. [2] A. Ananpinitwatna and T. Poomsa-ard, Hyperidentities in biregular leftmost graph varieties of type (2,0), Int. Math. Forum 4 (18) (2009), 845-864. Zbl1205.08004
  3. [3] K. Denecke and S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman and Hall/CRC 2002. 
  4. [4] K. Denecke and T. Poomsa-ard, Hyperidentities in graph algebras, Contributions to General Algebra and Aplications in Discrete Mathematics, Potsdam (1997), 59-68. Zbl0915.08004
  5. [5] K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, hyperequational classes and clone congruences, Verlag Hölder-Pichler-Tempsky, Wien, Contributions to General Algebra 7 (1991), 97-118. Zbl0759.08005
  6. [6] M. Krapeedang and T. Poomsa-ard, Biregular leftmost graph varieties of type (2,0), accepted to publish in AAMS. 
  7. [7] E.W. Kiss, R. Pöschel and P. Pröhle, Subvarieties of varieties generated by graph algebras, Acta Sci. Math. 54 (1990), 57-75. Zbl0713.08006
  8. [8] J.Khampakdee and T. Poomsa-ard, Hyperidentities in (xy)x ≈ x(yy) graph algebras of type (2,0), Bull. Khorean Math. Soc. 44 (4) (2007), 651-661. Zbl1142.08003
  9. [9] J. Płonka, Hyperidentities in some of vareties, pp. 195-213 in: General Algebra and discrete Mathematics ed. by K. Denecke and O. Lüders, Lemgo 1995. Zbl0813.08004
  10. [10] J. Płonka, Proper and inner hypersubstitutions of varieties, pp. 106-115 in: 'Proceedings of the International Conference: Summer School on General Algebra and Ordered Sets 1994', Palacký University Olomouce 1994. Zbl0828.08003
  11. [11] T. Poomsa-ard, Hyperidentities in associative graph algebras, Discussiones Mathematicae General Algebra and Applications 20 (2) (2000), 169-182. Zbl0977.08006
  12. [12] T. Poomsa-ard, J. Wetweerapong and C. Samartkoon, Hyperidentities in idempotent graph algebras, Thai Journal of Mathematics 2 (2004), 171-181. Zbl1066.05074
  13. [13] T. Poomsa-ard, J. Wetweerapong and C. Samartkoon, Hyperidentities in transitive graph algebras, Discussiones Mathematicae General Algebra and Applications 25 (1) (2005), 23-37. Zbl1102.08004
  14. [14] R. Pöschel, The equational logic for graph algebras, Zeitschr.f.math. Logik und Grundlagen d. Math. Bd. S. 35 (1989), 273-282. Zbl0661.03020
  15. [15] R. Pöschel, Graph algebras and graph varieties, Algebra Universalis 27 (1990), 559-577. Zbl0725.08002
  16. [16] R. Pöschel and W. Wessel, Classes of graph definable by graph algebras identities or quasiidentities, Comment. Math. Univ, Carolinae 28 (1987), 581-592. Zbl0621.05030
  17. [17] C.R. Shallon, Nonfinitely based finite algebras derived from lattices, Ph. D. Dissertation, Uni. of California, Los Angeles 1979. 

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