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Congruence Class Sizes in Finite Sectionally Complemented Lattices

  • G. Grätzer (a1) and E. T. Schmidt (a2)
Abstract

The congruences of a finite sectionally complemented lattice L are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence Θ of L is from being uniform, we introduce Spec Θ, the spectrum of Θ, the family of cardinalities of the congruence classes of Θ. A typical result of this paper characterizes the spectrum S = (mj | j < n) of a nontrivial congruence Θ with the following two properties:

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References
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[1] Grätzer, G., General Lattice Theory, Second edition: New appendices by the author with B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung, and R.Wille. Birkhäuser Verlag, Basel, 1998.
[2] Grätzer, G. and Schmidt, E. T., On congruence lattices of lattices. Acta Math. Acad. Sci. Hungar. 13 (1962), 179185.
[3] Grätzer, G. and Schmidt, E. T., Congruence-preserving extensions of finite lattices into sectionally complemented lattices. Proc. Amer. Math. Soc. 127 (1999), 19031915.
[4] Grätzer, G. and Schmidt, E. T., Finite lattices with isoform congruences. Tatra Mt. Math. Publ. 27 (2003), 114.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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