# Article

 Title: Ideal extensions of graph algebras  (English) Author: Čipková, Karla Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 Volume: 56 Issue: 3 Year: 2006 Pages: 933-947 Summary lang: English . Category: math . Summary: Let $\mathcal A$ and $\mathcal B$ be graph algebras. In this paper we present the notion of an ideal in a graph algebra and prove that an ideal extension of $\mathcal A$ by $\mathcal B$ always exists. We describe (up to isomorphism) all such extensions. Keyword: oriented graph Keyword: graph (Shallon) algebra Keyword: congruence relation Keyword: ideal Keyword: quotient graph algebra Keyword: ideal extension MSC: 08A30 idZBL: Zbl 1164.08300 idMR: MR2261665 . Date available: 2009-09-24T11:39:49Z Last updated: 2012-05-31 Stable URL: http://hdl.handle.net/10338.dmlcz/128118 . Reference: [1] A. H. Clifford: Extension of semigroup.Trans. Amer. Math. Soc. 68 (1950), 165–173. MR 0033836 Reference: [2] A. J. Hullin: Extension of ordered semigroup.Czechoslovak Math. J. 26(101) (1976), 1–12. Reference: [3] D. Jakubíková-Studenovská: Subalgebra extensions of partial monounary algebras.Czechoslovak Math. J, Submitted. MR 2261657 Reference: [4] N. Kehaypulu, P. Kiriakuli: The ideal extension of lattices.Simon Stevin 64, 51–56. MR 1072483 Reference: [5] N. Kehaypulu, M. Tsingelis: The ideal extension of ordered semigroups.Commun. Algebra 31 (2003), 4939–4969. MR 1998037 Reference: [6] E. W. Kiss, R. Pöschel, P. Pröhle: Subvarieties of varieties generated by graph algebras.Acta Sci. Math. 54 (1990), 57–75. MR 1073419 Reference: [7] J. Martinez: Torsion theory of lattice ordered groups.Czechoslovak Math. J. 25(100) (1975), 284–299. MR 0389705 Reference: [8] S. Oates-Macdonald, M. Vaughan-Lee: Varieties that make one cross.J. Austral. Math. Soc. (Ser. A) 26 (1978), 368–382. MR 0515754 Reference: [9] S. Oates-Williams: On the variety generated by Murskii’s algebra.Algebra Universalis 18 (1984), 175–177. Zbl 0542.08004, MR 0743465 Reference: [10] R. Pöschel: Graph algebras and graph varieties.Algebra Universalis 27 (1990), 559–577. MR 1387902 Reference: [11] R. Pöschel: Shallon algebras and varieties for graphs and relational systems.Algebra und Graphentheorie (Jahrestagung Algebra und Grenzgebiete), Bergakademie Freiberg, Section Math., Siebenlehn, 1986, pp. 53–56. Reference: [12] C. R. Shallon: Nonfinitely based finite algebras derived from lattices.PhD.  Dissertation, U.C.L.A, 1979. .

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