A note on r− S lemma


Abstract
Rauszer and Sabalski proved in [2] that distributivity with respect to infi- nite joins and meets is a sucient and necessary condition making the RasiowaSikorski Lemma valid in distributive lattices. The main part of their proof is a direct construction of a required filter under distributivity. In this note we show that a generalization of the result can be obtained from the Rasiowa-Sikorski Lemma for Boolean algebras by using Gornemann’s result in [1] instead of a direct con- ¨ struction. Suppose A is a distributive lattice and Q; R 2 A f;g. We call A complete if 8M 2 Q9 M 2 A and 8N 2 R9 F N 2 A. M 2 Q is -dis if 9 M 2 A and 8a 2 A a t M = m2M. N 2 R is F -dis if 9 F N 2 A and 8a 2 A a u F N = F n2N. is called distributive if every M 2 Q is -dis and every N 2 R is F -dis. Suppose A is complete and C; D A. By rC we mean the filter in A generated by C, in particular, let r; = ; = ;. is called complete if and , where 8M 2 Q 9m 2 MrC \ D[fmg = ;) 8N 2 R 9n 2 NrC[fng \ D = ;)
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