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oriented graph; graph (Shallon) algebra; congruence relation; ideal; quotient graph algebra; ideal extension
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.
[1] A. H. Clifford: Extension of semigroup. Trans. Amer. Math. Soc. 68 (1950), 165–173. DOI 10.1090/S0002-9947-1950-0033836-2 | MR 0033836
[2] A. J. Hullin: Extension of ordered semigroup. Czechoslovak Math. J. 26(101) (1976), 1–12.
[3] D. Jakubíková-Studenovská: Subalgebra extensions of partial monounary algebras. Czechoslovak Math. J, Submitted. MR 2261657
[4] N. Kehaypulu, P. Kiriakuli: The ideal extension of lattices. Simon Stevin 64, 51–56. MR 1072483
[5] N. Kehaypulu, M. Tsingelis: The ideal extension of ordered semigroups. Commun. Algebra 31 (2003), 4939–4969. DOI 10.1081/AGB-120023141 | MR 1998037
[6] E. W. Kiss, R. Pöschel, P. Pröhle: Subvarieties of varieties generated by graph algebras. Acta Sci. Math. 54 (1990), 57–75. MR 1073419
[7] J. Martinez: Torsion theory of lattice ordered groups. Czechoslovak Math. J. 25(100) (1975), 284–299. MR 0389705
[8] S. Oates-Macdonald, M. Vaughan-Lee: Varieties that make one cross. J. Austral. Math. Soc. (Ser. A) 26 (1978), 368–382. DOI 10.1017/S1446788700011897 | MR 0515754
[9] S. Oates-Williams: On the variety generated by Murskii’s algebra. Algebra Universalis 18 (1984), 175–177. DOI 10.1007/BF01198526 | MR 0743465 | Zbl 0542.08004
[10] R. Pöschel: Graph algebras and graph varieties. Algebra Universalis 27 (1990), 559–577. DOI 10.1007/BF01189000 | MR 1387902
[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.
[12] C. R. Shallon: Nonfinitely based finite algebras derived from lattices. PhD.  Dissertation, U.C.L.A, 1979.
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