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

Apinant Anantpinitwatna ; Tiang Poomsa-ard

Discussiones Mathematicae - General Algebra and Applications, Tome 29 (2009), p. 81-107 / Harvested from The Polish Digital Mathematics Library

Résumé

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].

Détail

Publié le : 2009-01-01

Zbl 1198.05086

EUDML-ID : urn:eudml:doc:276824

Bibliographie

[000] [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. | Zbl 1201.08006

[001] [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. | Zbl 1205.08004

[002] [3] K. Denecke and S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman and Hall/CRC 2002.

[003] [4] K. Denecke and T. Poomsa-ard, Hyperidentities in graph algebras, Contributions to General Algebra and Aplications in Discrete Mathematics, Potsdam (1997), 59-68. | Zbl 0915.08004

[004] [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. | Zbl 0759.08005

[005] [6] M. Krapeedang and T. Poomsa-ard, Biregular leftmost graph varieties of type (2,0), accepted to publish in AAMS.

[006] [7] E.W. Kiss, R. Pöschel and P. Pröhle, Subvarieties of varieties generated by graph algebras, Acta Sci. Math. 54 (1990), 57-75. | Zbl 0713.08006

[007] [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. | Zbl 1142.08003

[008] [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. | Zbl 0813.08004

[009] [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. | Zbl 0828.08003

[010] [11] T. Poomsa-ard, Hyperidentities in associative graph algebras, Discussiones Mathematicae General Algebra and Applications 20 (2) (2000), 169-182. | Zbl 0977.08006

[011] [12] T. Poomsa-ard, J. Wetweerapong and C. Samartkoon, Hyperidentities in idempotent graph algebras, Thai Journal of Mathematics 2 (2004), 171-181. | Zbl 1066.05074

[012] [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. | Zbl 1102.08004

[013] [14] R. Pöschel, The equational logic for graph algebras, Zeitschr.f.math. Logik und Grundlagen d. Math. Bd. S. 35 (1989), 273-282. | Zbl 0661.03020

[014] [15] R. Pöschel, Graph algebras and graph varieties, Algebra Universalis 27 (1990), 559-577. | Zbl 0725.08002

[015] [16] R. Pöschel and W. Wessel, Classes of graph definable by graph algebras identities or quasiidentities, Comment. Math. Univ, Carolinae 28 (1987), 581-592. | Zbl 0621.05030

[016] [17] C.R. Shallon, Nonfinitely based finite algebras derived from lattices, Ph. D. Dissertation, Uni. of California, Los Angeles 1979.