A CHARACTERIZATION OF MEET AND JOIN RESPECTING PRE-ORDERS AND CONGRUENCES ON FINITE LATTICES

A connection between pre-orders that respect the operations of the lattice and sets of join-irreducibles closed under a relation almost-equal-to between join-irreducibles is demonstrated. It is shown that any lattice pre-order determines two sets of join-irreducibles closed under the relation almost-equal-to and that elements of the lattice are related by the pre-order if and only if the subsets of join-irreducibles which they are greater than are comparable. The above connection is extended to congruences and the set of join-irreducibles that determine congruences that produce distributive quotient lattices are characterised. Finally it is shown that the quotient lattice of an arbitrary congruence is isomorphic to the lattice of decreasing subsets of join-irreducibles that determine the congruence