Ordered Relations, Concept Lattices and Self-bonds
Kalle Kaarli and Sándor Radeleczki
Valentini introduced ordered relations on complete lattices for representing unital quantales. Here we study interrelations between ordered relations, tolerances and formal contexts. We show that for any complete tolerance 𝑇 of a complete lattice 𝐿, the factor lattice 𝐿/𝑇 is iso- morphic to the concept lattice of the context (𝐿, 𝐿, 𝑅), where 𝑅 stands for the ordered relation ≤ ◦ 𝑇. We also prove that any ordered relation on a complete sublattice of 𝐿 can be extended to an ordered relation on 𝐿. Finally, we show that the self-bonds of an arbitrary formal context 𝐾 and the ordered relations on the concept lattice of this context 𝐾 form isomorphic lattices.
1991 Mathematics Subject Classiffication. Primary: 06B23, 06F07; Secondary: 06F07, 06A15, 06B15, 06B99
Keywords: Complete lattice, tolerance relation, ordered relation, quantale, formal context, concept lattice, self-bond