Hyperidentities in transitive graph algebras

Tiang Poomsa-ard; Jeerayut Wetweerapong; Charuchai Samartkoon

Discussiones Mathematicae - General Algebra and Applications (2005)

  • Volume: 25, Issue: 1, page 23-37
  • 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 an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G = (V,E) is called a transitive graph if the corresponding graph algebra A(G) satisfies the equation x(yz) ≈ (xz)(yz). An identity s ≈ t of terms s and t of any type t is called a hyperidentity of an algebra A̲ if whenever the operation symbols occurring in s and t are replaced by any term operations of A of the appropriate arity, the resulting identities hold in A̲ . In this paper we characterize transitive graph algebras, identities and hyperidentities in transitive graph algebras.

How to cite

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Tiang Poomsa-ard, Jeerayut Wetweerapong, and Charuchai Samartkoon. "Hyperidentities in transitive graph algebras." Discussiones Mathematicae - General Algebra and Applications 25.1 (2005): 23-37. <http://eudml.org/doc/287691>.

@article{TiangPoomsa2005,
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 an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G = (V,E) is called a transitive graph if the corresponding graph algebra A(G) satisfies the equation x(yz) ≈ (xz)(yz). An identity s ≈ t of terms s and t of any type t is called a hyperidentity of an algebra A̲ if whenever the operation symbols occurring in s and t are replaced by any term operations of A of the appropriate arity, the resulting identities hold in A̲ . In this paper we characterize transitive graph algebras, identities and hyperidentities in transitive graph algebras.},
author = {Tiang Poomsa-ard, Jeerayut Wetweerapong, Charuchai Samartkoon},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {identity; hyperidentity; term; normal form term; binary algebra; graph algebra; transitive graph algebra},
language = {eng},
number = {1},
pages = {23-37},
title = {Hyperidentities in transitive graph algebras},
url = {http://eudml.org/doc/287691},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Tiang Poomsa-ard
AU - Jeerayut Wetweerapong
AU - Charuchai Samartkoon
TI - Hyperidentities in transitive graph algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2005
VL - 25
IS - 1
SP - 23
EP - 37
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 an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G = (V,E) is called a transitive graph if the corresponding graph algebra A(G) satisfies the equation x(yz) ≈ (xz)(yz). An identity s ≈ t of terms s and t of any type t is called a hyperidentity of an algebra A̲ if whenever the operation symbols occurring in s and t are replaced by any term operations of A of the appropriate arity, the resulting identities hold in A̲ . In this paper we characterize transitive graph algebras, identities and hyperidentities in transitive graph algebras.
LA - eng
KW - identity; hyperidentity; term; normal form term; binary algebra; graph algebra; transitive graph algebra
UR - http://eudml.org/doc/287691
ER -

References

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  1. [1] K. Denecke and T. Poomsa-ard, Hyperidentities in graph algebras, 'Contributions to General Algebra and Applications in Discrete Mathematics', Shaker-Verlag, Aachen 1997, 59-68. Zbl0915.08004
  2. [2] K. Denecke and M. Reichel, Monoids of hypersubstitutions and M-solid varieties, 'Contributions to General Algebra', vol. 9, Verlag Hölder-Pichler-Tempsky, Vienna 1995, 117-125. Zbl0884.08008
  3. [3] E.W. Kiss, R. Pöschel and P. Pröhle, Subvarieties of varieties generated by graph algebras, Acta Sci. Math. (Szeged) 54 (1990), 57-75. Zbl0713.08006
  4. [4] J. Płonka, Hyperidentities in some of vareties, 'General Algebra and Discrete Mathematics', Heldermann Verlag, Lemgo 1995, 195-213. 
  5. [5] J. Płonka, Proper and inner hypersubstitutions of varieties, 'Proceedings of the International Conference: 'Summer School on General Algebra and Ordered Sets', Olomouc 1994', Palacký University, Olomouc 1994, 106-115. Zbl0828.08003
  6. [6] T. Poomsa-ard, Hyperidentities in associative graph algebras, Discuss. Math. - Gen. Algebra Appl. 20 (2000), 169-182. Zbl0977.08006
  7. [7] R. Pöschel, The equational logic for graph algebras, Z. Math. Logik Grundl. Math. 35 (1989), 273-282. Zbl0661.03020
  8. [8] R. Pöschel, Graph algebras and graph varieties, Algebra Universalis 27 (1990), 559-577. Zbl0725.08002
  9. [9] C.R. Shallon, Nonfinitely Based Finite Algebras Derived from Lattices, Ph.D. Thesis, University of California, Los Angeles, CA, 1979. 

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