Chain conditions in free products of lattices with infinitary ... - MSP

CHAIN CONDITIONS IN FREE PRODUCTS OF LATTICES 113
be an m-homomorphism such that ψ(β) = 0 and ψ(β + 1) = 1. We
now define the m-homomorphism φ: L —• B t ϋ {0,1} by φ(x) if x e B 19
and φ(x) = ψ(x) if xe B 2. Since φ((x Vβ) Λ (β + 1)) = #, it now
follows that I£7/1 = n β. Therefore, \C\ = n, completing the proof.
Theorem 4 is easier to prove. Indeed, if n (ajσ( i), , #„(#!, ••-,!/») be proper representations with « V%} We can assume there is an integer k with 0 ^ k