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Related structures with involution

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Abstract

We give a necessary and sufficient condition for an involution lattice to be isomorphic to the direct square of its invariant part. This result is applied to show relations between related lattices of an algebra. For instance, generalizing some earlier results of G. Czédli and L. Szabó it is proved that any algebra admits a connected compatible partial order whenever its quasiorder lattice is isomorphic to the direct square of its congruence lattice. Further, a majority algebra is lattice ordered if and only if the lattice of its compatible reflexive relations is isomorphic to the direct square of its tolerance lattice. In the latter case, one can establish a bijective correspondence between factor congruence pairs of the algebra and its pairs of compatible lattice orders; several consequences of this result are given.

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Authors and Affiliations

  1. Institut für Algebra, TU Dresden, 01062, Dresden, Germany

    R. Pöschel

  2. Institute of Mathematics, University of Miskolc, 3515, Miskolc-Egyetemváros, Hungary

    S. Radeleczki

Authors
  1. R. Pöschel
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  2. S. Radeleczki
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Correspondence to R. Pöschel.

Additional information

This paper is a result of a collaboration of the authors within DFG-MTA Grant no. 167, whose support is gratefully acknowledged by the authors. The partial support by Hungarian National Research Fund (Grant no. T049433/05) is also acknowledged by the second author.

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Pöschel, R., Radeleczki, S. Related structures with involution. Acta Math Hung 123, 169–185 (2009). https://doi.org/10.1007/s10474-009-8081-6

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  • DOI: https://doi.org/10.1007/s10474-009-8081-6

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