Modular varieties with the Fraser-Horn property

Author:
Diego Vaggione

Journal:
Proc. Amer. Math. Soc. **127** (1999), 701-708

MSC (1991):
Primary 08A05, 08B10

DOI:
https://doi.org/10.1090/S0002-9939-99-04647-X

MathSciNet review:
1473681

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Abstract | References | Similar Articles | Additional Information

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,*with factorable congruences and**imply**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**

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Additional Information

**Diego Vaggione**

Affiliation:
Facultad de Matemática, Astronomía y Física (Fa.M.A.F.), Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba 5000, Argentina

Email:
vaggione@mate.uncor.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-04647-X

Received by editor(s):
April 24, 1997

Received by editor(s) in revised form:
July 7, 1997

Additional Notes:
This research was supported by CONICOR and SECYT (UNC)

Communicated by:
Carl Jockusch

Article copyright:
© Copyright 1999
American Mathematical Society