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  • 1 TU Dresden Institut für Algebra 01062 Dresden Germany
  • | 2 University of Miskolc Institute of Mathematics 3515 Miskolc-Egyetemváros Hungary
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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.