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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Varieties with definable factor congruences


Authors: Pedro Sánchez Terraf and Diego J. Vaggione
Journal: Trans. Amer. Math. Soc. 361 (2009), 5061-5088
MSC (2000): Primary 08B05; Secondary 03C40
Published electronically: May 18, 2009
MathSciNet review: 2515803
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Abstract: We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that $ \mathcal{V}$ has DFC if and only if $ \mathcal{V}$ has $ \vec{0}$ & $ \vec{1}$ and Boolean Factor Congruences. We also obtain an explicit first-order definition $ \Phi$ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner that it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.


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Additional Information

Pedro Sánchez Terraf
Affiliation: CIEM — Facultad de Matemática, Astronomía y Física (Fa.M.A.F.), Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba 5000, Argentina
Email: sterraf@famaf.unc.edu.ar

Diego J. Vaggione
Affiliation: CIEM — 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: http://dx.doi.org/10.1090/S0002-9947-09-04921-6
PII: S 0002-9947(09)04921-6
Received by editor(s): December 15, 2006
Published electronically: May 18, 2009
Additional Notes: This work was supported by CONICET and SECYT-UNC
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.