Abstract
We say that a variety V of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every A ∈ V is closed under intersection. We investigate the congruence lattices of algebras in locally finite, congruence-distributive CIP varieties and obtain a complete characterization for several types of such varieties. It turns out that our description only depends on subdirectly irreducible algebras in V and embeddings between them. We believe that the strategy used here can be further developed and used to describe the congruence lattices for any (locally finite) congruence-distributive CIP variety.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Agliano, K. A. Baker: Congruence intersection properties for varieties of algebras. J. Aust. Math. Soc., Ser. A 67 (1999), 104–121.
K. A. Baker: Primitive satisfaction and equational problems for lattices and other algebras. Trans. Am. Math. Soc. 190 (1974), 125–150.
W. J. Blok, D. Pigozzi: On the structure of varieties with equationally definable principal congruences. I. Algebra Univers. 15 (1982), 195–227.
P. Gillibert, M. Ploščica: Congruence FD-maximal varieties of algebras. Int. J. Algebra Comput. 22 (2012), 1250053, 14 pages.
G. Gratzer: General Lattice Theory. (With appendices with B. A. Davey, R. Freese, B. Ganter, et al.) Paperback reprint of the 1998 2nd edition, Birkhauser, Basel, 2003.
T. Katriňák: A new proof of the construction theorem for Stone algebras. Proc. Am. Math. Soc. 40 (1973), 75–78.
T. Katriňák, A. Mitschke: Stonesche Verbande der Ordnung n und Postalgebren. Math. Ann. 199 (1972), 13–30. (In German.)
K. B. Lee: Equational classes of distributive pseudo-complemented lattices. Can. J. Math. 22 (1970), 881–891.
M. Ploščica: Separation in distributive congruence lattices. Algebra Univers. 49 (2003), 1–12.
M. Ploščica: Finite congruence lattices in congruence distributive varieties. Proceedings of the 64th workshop on general algebra “64. Arbeitstagung Allgemeine Algebra”, Olomouc, Czech Republic, May 30–June 2, 2002 and of the 65th workshop on general algebra “65. Arbeitstagung Allgemeine Algebra”, Potsdam, Germany, March 21–23, 2003 (I. Chajda, et al., ed.). Verlag Johannes Heyn, Klagenfurt. Contrib. Gen. Algebra 14, 2004, pp. 119–125.
M. Ploščica: Local separation in distributive semilattices. Algebra Univers. 54 (2005), 323–335.
M. Ploščica, J. Tůma, F. Wehrung: Congruence lattices of free lattices in non-distributive varieties. Colloq. Math. 76 (1998), 269–278.
P. Růžička: Lattices of two-sided ideals of locally matricial algebras and the Γ-invariant problem. Isr. J. Math. 142 (2004), 1–28.
E. T. Schmidt: The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice. Acta Sci. Math. (Szeged) 43 (1981), 153–168.
F. Wehrung: A uniform refinement property for congruence lattices. Proc. Am. Math. Soc. 127 (1999), 363–370.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by VEGA Grant 2/0194/10 and VVGS-PF-2012-50
Rights and permissions
About this article
Cite this article
Krajník, F., Ploščica, M. Congruence lattices in varieties with compact intersection property. Czech Math J 64, 115–132 (2014). https://doi.org/10.1007/s10587-014-0088-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-014-0088-7