G. N. Raney. Tight Galois connections and complete distributivity. Trans. AMS, 97:418--426, 1960.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....notion of nuclearity in the category of complete join semilattices CJSL. It is well known that this is a symmetric monoidal closed category, in fact autonomous [41] The authors completely characterize nuclearity in this case (This result is closely related to Raney s notion of a tight morphism [48]. Theorem 5.5 (Higgs, Rowe) A morphism f : A B in CJSL is nuclear if and only if there exists g: B A such that for all a 2 A; f(a) supfbja 6 g(b)g. An object is nuclear if and only if it is completely distributive. Remark 5.6 Following recent work of Joyal, Street and Verity [40] on ....

G. Raney. Tight Galois connections and complete distributivity. Transactions of the American Mathematical Society, 97:418--426, (1960)

....L (id L ) P L ( fx y: y x in Lg) fP L (x y) y x in Lg = fz y: y x; z = L n (x) g Each of the functions z y has at most two points in its image, so the image is certainly completely distributive. Therefore, each of these functions is a tight Galois Connection on L [16]. The supremum of tight Galois Connections is tight, for every Galois Connection has a least tight Galois Connection below it. Thus id L is tight as well. By [16] this is the case if and only if L is completely distributive. 2 Hence we can make W fP L (x y) y x in Lg a precise measure of ....

....in its image, so the image is certainly completely distributive. Therefore, each of these functions is a tight Galois Connection on L [16] The supremum of tight Galois Connections is tight, for every Galois Connection has a least tight Galois Connection below it. Thus id L is tight as well. By [16] this is the case if and only if L is completely distributive. 2 Hence we can make W fP L (x y) y x in Lg a precise measure of complete distributivity. Corollary 4 Let L be a continuous lattice. Then the following are equivalent: 1. L is completely distributive. 2. id L = W fP L (x ....

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G. N. Raney. Tight Galois connections and complete distributivity. Trans. AMS, 97:418--426, 1960.

....The domain logic is sound and complete. As a formula: 8M; Gamma; OE: Gamma M : OE if and only if Gamma ffl M : OE : Exercises 7.3.19. 1. Prove that a completely distributive lattice also satisfies the dual distributivity axiom: W i2I V A i = V f :I fi Gamma [A i W i2I f(i) 2. [Raney, 1960] Prove that a complete lattice L is completely distributive if and only if the following holds for all x 2 L: x = a6x b6a b : Hint: Use Theorem 7.1.3. 3. Show that a topological space is sober if and only if every irreducible closed set is the closure of a unique point. 4. Find a complete ....

G. N. Raney. Tight Galois connections and complete distributivity. Trans. AMS, 97:418--426, 1960.

....which is useful in the semantics of concurrent processes where one needs a multi cut rule [Abr93] to model cyclic networks: CD is compact closed [Bar79] i.e. A Omega B = AOB (4) is a natural isomorphism in CD. The proof of that uses the concept of tight Galois Connections introduced in [Ran60] A Galois Connection between complete lattices L and M is a pair of antitone ( order reversing) maps r: L M and n: M L such that n(r(x) x and r(n(y) y for all x 2 L and all y 2 M . If L or M is completely distributive, then every Galois Connection [r; n] between L and M has a canonical ....

....L and M has a canonical form as follows. Definition 1 For complete lattices L and M and L Theta M , define : fhl; mi 2 L Theta M j 8hu; vi 2 : l u or m vg (5) fhl; mi 2 L Theta M j 8hu; vi 2 : l u or m vg: A Galois Connection [r; n] between L and M is tight [Ran60] iff there exists some L Theta M such that for all x 2 L and all y 2 M we have r(x) fy j hx; yi 2 g (6) n(y) fx j hx; yi 2 g: Clearly, we can identify tight Galois Connections between L and M with subsets L Theta M . We get a dual picture if we consider subsets ....

[Article contains additional citation context not shown here]

G. N. Raney. Tight Galois connections and complete distributivity. Trans. AMS, 97:418--426, 1960.

....that we get approximation from both sides automatically in super continuous lattices. Observe, however, that the relations nA and (nA op) Gamma1 are different in general. We will also make use of the following observation which is a consequence of Raney s work on tight Galois connections, Ran60] Theorem 19 (Raney) A complete lattice A is completely distributive if and only if for every a 2 A we have a = V a 0 6a W a 00 6a 0 a 00 . Proof. if : It is easy to see that for every a 0 6 a the element x : W a 00 6a 0 a 00 is completely above a. Hence A op is ....

G. N. Raney. Tight Galois connections and complete distributivity. Trans. AMS, 97:418--426, 1960.

....Theorem 7.3.18. The domain logic is sound and complete. As a formula: 8M;0; OE: 0 M : OE if and only if 0 M : OE : Exercises 7.3.19. 1. Prove that a completely distributive lattice also satisfies the dual distributivity axiom: W i2I V A i = V f :I fi 0 [A i W i2I f(i) 2. [Raney, 1960] Prove that a complete lattice L is completely distributive if and only if the following holds for all x 2 L: x = a6x b6a b : Hint: Use Theorem 7.1.3. 3. Show that a topological space is sober if and only if every irreducible closed set is the closure of a unique point. 4. Find a complete ....

G. N. Raney. Tight Galois connections and complete distributivity. Trans. AMS, 97:418--426, 1960.

....ut The way way below relation on a complete lattice is defined analogously to the way below relation. The difference is that one refers to arbitrary suprema instead of directed suprema. A complete lattice is completely distributive if every element is the supremum of all elements way way below it [11, 12]. If L is completely distributive then we obtain similarly that P L (X) is a supprojection of [O(X) L] and the way way below relation on P L (X) is induced by the way way below relation on [O(X) L] Let us also point out that for continuous X and completely distributive lattices L our ....

G. N. Raney. Tight Galois connections and complete distributivity. Trans. AMS, 97:418--426, 1960.

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G. N. Raney. Tight Galois connections and complete distributivity. Trans. AMS, 97:418--426, 1960.

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