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.
We prove that the tolerance lattice TolA of an algebra A from a congruence modular variety V is 0-1 modular and satisfies the general disjointness property. If V is congruence distributive, then the lattice Tol A is pseudocomplemented. If V admits a majority term, then Tol A is 0-modular.