The order of principal congruences of a bounded lattice (Q364684)

Using the new terminology introduced by the author, the words `poset', `partially ordered set' and `order' are synonyms (see the title). The paper under review deals with the representation of lattices by congruence lattices of lattices. Since congruence lattices Con\(L\) are algebraic and distributive, Con\(L\) is uniquely determined by its compact elements Con\(_cL\). It is known that Con\(_cL\) is a join-semilattice with zero. Moreover, Con\(_cL\) is again determined by Princ\(L\), the poset of all principal congruences on \(L\). (Note that Princ\(L\) is join-generating Con\(_cL\).) Clearly, we can ask after a characterization of the order Princ\(L\). It is easy to verify that Princ\(L\) is up-directed and with zero. The author gives an answer for bounded lattices: Let \(P\) be an (arbitrary) order with zero and unit. Then there exists a bounded lattice \(L\) such that \(P\cong \text{Princ}L\). If \(P\) is finite, then \(L\) can be constructed finite as well. In addition, the author presents 14 open problems concerning the representation of lattices by congruence lattices of lattices.