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

110 G. GRATZER, A. HAJNAL AND DAVID KELLY
Hence, we can assume that each element of Y has a proper representation
a = p(ά), where the same m-polynomial p is used for each
element of Y. For notational simplicity, we further assume that,
for some cardinal m < m, a — (x 0 a
a \ a < m> whenever ae Y, where
o
tfel for all a < m . (Note that x 0 a
Φ xa for a Φ β.)
a β
Consider the sets S = {x a a
a \ a < m } for a e Γ. By the Erdόs-Rado
0
theorem, there is a subset 3Γ' £ 3f with | Y' \ — n such that (Sa\a e
Y f
) is a J-system, whose kernel we denote by D. For each ae Y',
the inclusion D Q Sa induces a map >fra: D-+ m0 in the obvious way.
Since \{fa\ae Y'}\