# Alpha-cut-complete Boolean algebras by Hager A.W. PDF

By Hager A.W.

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It seems reasonable to begin with semilattices as examples of non-congruencepermuting algebras. l {(x,y) E AxA : x¢> = YtP} for some pair of surjective morphisms ¢>,tP: A -+ B. This necessary condition also suffices to make A congruence-permuting [4; see also 5, pp 125-127]. e. l (c;. may be replaced by equality here); and (iii) full. l are congruences on A corresponding to the morphisms ¢> and tP respectively and so I propose to call a relation satisfying (i), (ii) and (iii) above a bicongruence on A, and to notate the set of such by ~e(A).

Every finitely generated left ideal of S is FD. (a) For every xES, the principal left ideal Sx is FD, and (b) For every x, YES, SxnSy is either a principal left ideal or empty. 31 Proof: The equivalence of (1) and (2) is immediate from Lemma I, Definition 3, and the fact that the criterion for weak flatness could have equivalently been stated in terms of finitely generated left ideals. and Assuming (2), 3(a) is obvious. To obtain 3(b) suppose x, yES Sx n Sy =I 0. Let r index the set {(s, t) E S x S I sx = ty} = {(s" t,) I, E r}.

5. For a complete lattice L, the following conditions are equivalent: (a) o:AVB = V{o:*b : bE B