Related structures with involution (Q1046877)

The authors give necessary and sufficient condition for a bounded involution lattice to be isomorphic to the direct square of its invariant part. Using this result, they show relations between related lattices of an algebra. 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, the authors show that a majority algebra is lattice-ordered if the lattice of its compatible reflexive relations is isomorphic to the direct square of its tolerance lattice. The bijective correspondence is established between factor congruence pairs of the algebra and its pairs of compatible lattice orders.