Weak Congruence Representability of Suborders and Direct Products

Here we treat the problem of representability of an algebraic lattice by the weak congruence lattice of an algebra, i.e. the lattice of all symmetric and transitive relations compatible with algebra. We prove that some suborders of the representable lattices are representable, and give conditions under which these suborders are also sublattices of the initial lattices. We also prove that the direct product of a set of representable lattices, slightly extended, is representable itself.