G. Cz\xe9dli\u2019s tolerance factor lattice construction and weak ordered relations

Abstract

G. Czédli proved that the blocks of any compatible tolerance T of a lattice L can be ordered in such a way that they form a lattice L/T called the factor lattice of L modulo T. Here we show that the Dedekind–MacNeille completion of the lattice L/T is isomorphic to the concept lattice of the context (L, L, R), where R stands for the reflexive weak ordered relation mathord {le } circ T. Weak ordered relations constitute the generalization of the ordered relations introduced by S. Valentini. Reflexive weak ordered relations can be characterized as compatible reflexive relations Rsubseteq L^{2} satisfying R= mathord {le } circ Rcirc mathord {le } .