Theorem 1. Let L be a finite lattice. Let δ be an equivalence relation on L with
intervals as equivalence classes. Then δ is a congruence relation iff the following
condition and its dual hold:
(C+) If x is covered by y, z ∈ L and x ≡ y (mod δ), then z ≡ y + z (mod δ).
2. The proof
We prove the join-substitution property: if x ≤ y and x ≡ y (mod δ), then
(1) x + z ≡ y + z (mod δ).
Let U = [x, y + z]. We induct on lengthU, the length of U.
\cite{Eubanks2015}
Let I = [y1, y + z] and J = [z1, y + z]. Then length I and lengthJ < lengthU.
Hence, the induction hypothesis applies to I and δI, and we obtain that w ≡ y+w
(mod δ). By the transitivity of δ, we conclude that
(2) z1 ≡ y + w (mod δ).
Therefore, applying the induction hypothesis to J and δJ, we conclude (1).
References
[1] G. Cz´edli, Patch extensions and trajectory colorings of slim rectangular lattices. Algebra Universalis
88 (2013), 255–280.
[2] G. Gra¨tzer, Congruences of fork extensions of lattices. Acta Sci. Math. (Szeged), 57 (2014),
417–434.
Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
Date: March 21, 2014.
2010 Mathematics Subject Classification. Primary: 06B10.
Key words and phrases. finite lattice, congruence.