By Pianigiani G.

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Ii (1) if and only if p = z X orp has is minimal, (2) (3) for any meet component r of pj, r or the duals of(1), (2) and (3) hold for the meet components ofp. ,P7e. All z but A(q)A(p), which shows that p is not minimal. ) Conversely, suppose p satisfies (1), (2) and (3), and let q be a minimal term such that = p. We want to show that A(p) = A(q), then p is also minimal. First observe that then q must have join components, for if q X or if q has meet components Qi,. . J = p we must have equality throughout, which contradicts the miimality of q.

Hence fl con(u, v) is the unique dual cover of con(u, v), and it follows that con(u, v) J(con(L)). To prove (ii) we simply choose p minimal with respect to the condition u = v+p. Then p J(L) and p/ps u/v. (lii) Supposeforsomeq J(L),q p, u/v. Thenu = v+p = v+q, / and bysemidistributivity u= v+pq. But thenpq v,henceu= v+pq= v, whichisacontradiction. pq / 0 From (i) and (ii) we conclude that for any finite lattice L the map p from J(L) to J(Ccn(L)) is onto. Day [79] shows that the map is one-one if and only if L is lower bounded.

By hypothesis a5. (in — 1,vi) as required. (in,vi) holds for all in,vi. 32 (Jónsson and Kiefer [62]). A finite lattice satisfies every clement has a canonical join representation. L, then u = 0 if and only if Let L be a finite lattice that satisfies (SD+), and suppose that I' and Ii' are two join representations of u L. By the preceding lemma the set {ab: a V, 6 W} is CBAJ'TLJ 36 c;LIVERjL 2. ifS again a join representation of a, and it clearly refines both and Since L is finite, a has oniy finitely nianv distinct join representations.

### A Baire category approach in existence theory of differential equations by Pianigiani G.

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