Proceedings of the American Mathematical Society

Author(s): Diego Vaggione
Journal: Proc. Amer. Math. Soc. 127 (1999), 701-708.
MSC (1991): Primary 08A05, 08B10
Retrieve article in: PDF
This article is available free of charge

Abstract: The notion of central idempotent elements in a ring can be easily generalized to the setting of any variety with the property that proper subalgebras are always nontrivial. We will prove that if such a variety is also congruence modular, then it has factorable congruences, i.e., it has the Fraser-Horn property. (This property is well known to have major implications for the structure theory of the algebras in the variety.)

1.: J. BERMAN and W. A. BLOK, The Fraser-Horn and apple properties, Trans. Amer. Math. Soc. 302 (1987), 427-465. MR 88h:08010
2.: D. BIGELOW and S. BURRIS, Boolean algebras of factor congruences, Acta Sci. Math., 54 (1990), 11-20. MR 91j:08012
3.: S. BURRIS, Remarks on the Fraser-Horn property, Algebra Universalis, 23 (1986), 19-21. MR 88b:08009
4.: S. BURRIS and H. SANKAPPANAVAR, A Course in Universal Algebra, Springer-Verlag, New York, 1981. MR 83k:08001
5.: C. C. CHANG and H. J. KEISLER, Model Theory, North-Holland, Amsterdam-London, 1973. MR 53:12927
6.: A. DAY, A characterization of modularity for congruence lattices of algebras, Canad. Math. Bull. 12 (1969), 167-173. MR 40:1317
7.: J. DUDA, Fraser-Horn identities can be written in two variables, Algebra Universalis, 26 (1989), 178-180. MR 90d:08009
8.: G. A. FRASER and A. HORN, Congruence relations in direct products, Proc. Amer. Math. 26 (1970), 390-394. MR 42:169
9.: G. GRÄTZER, General Lattice Theory, Birkhäuser Verlag, Basel and Stuttgart, 1978. MR 80c:06001a
10.: H. P. GUMM, Geometrical Methods in Congruence Modular Algebras, Mem. of the Amer. Math. Soc., 286 (1983). MR 85e:08012
11.: R. McKENZIE, G. McNULTY and W. TAYLOR, Algebras, Lattices, Varieties, Volume 1, The Wadsworth & Brooks/Cole Math. Series, Monterey, California (1987). MR 88e:08001
12.: J. KOLLAR, Congruences and one element subalgebras, Algebra Universalis, 9 (1979) 266-267. MR 80d:08011
13.: R. S. PIERCE, Modules over commutative regular rings, Mem. of the Amer. Math. Soc., 70 (1967). MR 36:151
14.: H. RIEDEL, Existentially closed algebras and Boolean products, Journal of Symbolic Logic 53 (1988), 571-596.
15.: D. VAGGIONE, ${\cal V}$ with factorable congruences and ${\cal V=}I\Gamma ^a({\cal V}_{DI})$ imply ${\cal V}$ is a discriminator variety, Acta Sci. Math. 62 (1996), 359-368. MR 97h:08012
16.: D. VAGGIONE, Varieties in which the Pierce stalks are directly indecomposable, Journal of Algebra 184 (1996), 424-434. MR 97f:08010
17.: D. VAGGIONE, Varieties of shells, Algebra Universalis, 36 (1996) 483-487. MR 97h:08004

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 08A05, 08B10

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS