By R. Keith Dennis

ISBN-10: 0387119663

ISBN-13: 9780387119663

ISBN-10: 3540119663

ISBN-13: 9783540119661

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Extra info for Algebraic K-Theory: Proceedings, Oberwolfach, FRG 1980

Example text

Q Q Accordingly the order-closed subalgebra of A generated by I is included in B and therefore in A. 313H Definitions It is worth distinguishing various types of supremum- and infimum-preserving function. Once again, I do this in almost the widest possible context. Let P and Q be two partially ordered sets, and φ : P → Q an order-preserving function, that is, a function such that φ(p) ≤ φ(q) in Q whenever p ≤ q in P . (a) I say that φ is order-continuous if (i) φ(sup A) = supp∈A φ(p) whenever A is a non-empty upwardsdirected subset of P and sup A is defined in P (ii) φ(inf A) = inf p∈A φ(p) whenever A is a non-empty downwards-directed subset of P and inf A is defined in P .

Suppose, if possible, that H ⊆ W is a non-empty open set and int φ[H] = ∅. Let b ∈ B \ {0} be such that b ⊆ H. Then φ[b] has empty interior; but also it is a closed set, so its complement is dense. Set A = {a : a ∈ A, a ∩ φ[b] = ∅}. Then a∈A a = Z \ φ[b] is a dense open set, so sup A = 1 in A (313Ca). Because π is order-continuous, sup π[A] = 1 in B (313L(b-iii)), and there is an a ∈ A such that πa ∩ b = 0. But this means that b ∩ φ−1 [a] = ∅ and φ[b] ∩ a = ∅, contrary to the definition of A. X X Thus there is no such set H, and (iii) is true.

G) Let P and Q be partially ordered sets, and φ : P → Q an order-preserving function. Show that φ is sequentially order-continuous iff φ−1 [C] is sequentially order-closed in A for every sequentially order-closed C ⊆ B. (h) For partially ordered sets P and Q, let us call a function φ : P → Q monotonic if it is either order-preserving or order-reversing. State and prove definitions and results corresponding to 313H, 313I and 313Xg for general monotonic functions. >(i) Let A be a Boolean algebra.