Article

 Title: Congruence lattices in varieties with compact intersection property (English) Author: Krajník, Filip Author: Ploščica, Miroslav Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 64 Issue: 1 Year: 2014 Pages: 115-132 Summary lang: English . Category: math . Summary: We say that a variety ${\mathcal V}$ of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every $A\in {\mathcal V}$ 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 ${\mathcal V}$ 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. (English) Keyword: compact congruence Keyword: congruence-distributive variety MSC: 06D15 MSC: 08A30 MSC: 08B10 idZBL: Zbl 06391481 idMR: MR3247449 DOI: 10.1007/s10587-014-0088-7 . Date available: 2014-09-29T09:42:07Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/143954 . Reference: [1] Agliano, P., Baker, K. A.: Congruence intersection properties for varieties of algebras.J. Aust. Math. Soc., Ser. A 67 (1999), 104-121. Zbl 0951.08003, MR 1699158, 10.1017/S1446788700000896 Reference: [2] Baker, K. A.: Primitive satisfaction and equational problems for lattices and other algebras.Trans. Am. Math. Soc. 190 (1974), 125-150. Zbl 0291.08001, MR 0349532, 10.1090/S0002-9947-1974-0349532-4 Reference: [3] Blok, W. J., Pigozzi, D.: On the structure of varieties with equationally definable principal congruences. I.Algebra Univers. 15 (1982), 195-227. Zbl 0512.08002, MR 0686803, 10.1007/BF02483723 Reference: [4] Gillibert, P., Ploščica, M.: Congruence FD-maximal varieties of algebras.Int. J. Algebra Comput. 22 1250053, 14 pages (2012). Zbl 1261.06013, MR 2974107, 10.1142/S0218196712500531 Reference: [5] Grätzer, G.: General Lattice Theory.(With appendices with B. A. Davey, R. Freese, B. Ganter, et al.) Paperback reprint of the 1998 2nd edition Birkhäuser, Basel (2003). Zbl 1152.06300, MR 1670580 Reference: [6] Katriňák, T.: A new proof of the construction theorem for Stone algebras.Proc. Am. Math. Soc. 40 (1973), 75-78. Zbl 0258.06006, MR 0316335, 10.2307/2038636 Reference: [7] Katriňák, T., Mitschke, A.: Stonesche Verbände der Ordnung $n$ und Postalgebren.Math. Ann. 199 (1972), 13-30 German. Zbl 0253.06009, MR 0319838, 10.1007/BF01419572 Reference: [8] Lee, K. B.: Equational classes of distributive pseudo-complemented lattices.Can. J. Math. 22 (1970), 881-891. Zbl 0244.06009, MR 0265240, 10.4153/CJM-1970-101-4 Reference: [9] Ploščica, M.: Separation in distributive congruence lattices.Algebra Univers. 49 (2003), 1-12. Zbl 1090.08003, MR 1978609 Reference: [10] Ploščica, M.: Finite congruence lattices in congruence distributive varieties.I. Chajda, et al. 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 Verlag Johannes Heyn, Klagenfurt. Contrib. Gen. Algebra {\it 14} 119-125 (2004). Zbl 1047.08005, MR 2059570 Reference: [11] Ploščica, M.: Local separation in distributive semilattices.Algebra Univers. 54 (2005), 323-335. Zbl 1086.06003, MR 2219414, 10.1007/s00012-005-1949-6 Reference: [12] Ploščica, M., Tůma, J., Wehrung, F.: Congruence lattices of free lattices in non-distributive varieties.Colloq. Math. 76 (1998), 269-278. Zbl 0904.06005, MR 1618712 Reference: [13] Růžička, P.: Lattices of two-sided ideals of locally matricial algebras and the $\Gamma$-invariant problem.Isr. J. Math. 142 (2004), 1-28. Zbl 1057.06004, MR 2085708, 10.1007/BF02771525 Reference: [14] Schmidt, E. T.: The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice.Acta Sci. Math. (Szeged) 43 (1981), 153-168. Zbl 0463.06007, MR 0621367 Reference: [15] Wehrung, F.: A uniform refinement property for congruence lattices.Proc. Am. Math. Soc. 127 (1999), 363-370. Zbl 0902.06006, MR 1468207, 10.1090/S0002-9939-99-04558-X .

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