Topics of divisibility
Bruno Bosbach · Kobra (Universitätsbibliothek Kassel) · 2015
∨ -complete extension of S if A ⊆ B =⇒ A | B is satisfied.Here | means right/left divisibility.But, in any case, each of these extensions provides a complete sup-latticemonoid, which is ∧ -distributive in the v-case and ∨ -distributive in the ucase.PROOF.This follows from the preceding results by 7.1.5. POne way of characterizing classes of lattices is the method of forbidden sublattices.The most trivial example: A lattice is a chain, if it does not contain any ({a, b, 0, 1}, ∧, ∨) with a ̸ ≤ b ̸ ≤ a and a ∧ b = 0, a ∨ b = 1 .Distributive and modular lattices admit a characterization by forbidding the subsequent lattices, called pentagon and diamond, respectively.