Chain conditions in free products of lattices with infinitary ... - MSP
Chain conditions in free products of lattices with infinitary ... - MSP
Chain conditions in free products of lattices with infinitary ... - MSP
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114 G. GRATZER, A. HAJNAL AND DAVID KELLY<br />
Let F be a nontrivial variety <strong>of</strong> fc^-<strong>lattices</strong> and let L be a F<strong>free</strong><br />
lattice generated by an <strong>in</strong>f<strong>in</strong>ite set X. We show that, <strong>in</strong><br />
contrast <strong>with</strong> the m = ^ 0 case, permutations <strong>of</strong> X can create dist<strong>in</strong>ct<br />
comparable elements <strong>in</strong> L. Let p and q be ^-polynomials <strong>in</strong><br />
variables {x n \ n < ft)} such that p ^ q holds <strong>in</strong> F (for any substitution)<br />
but p = q does not (for example, x 0 and x 0 VΛΓJ. Let x\ be<br />
dist<strong>in</strong>ct elements <strong>of</strong> X for is Z (the <strong>in</strong>tegers) and n < ω. Further,<br />
let Pi = p(x\\n
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