Finding paths through narrow and wide trees

We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.

THE EXISTENCE OF CERTAIN (CO-)LIMITS IN THE CATEGORY OF CHAIN FUNCTORS

Chain functors A*, B* have been introduced for calculating generalized homology theories by using chains and cycles, as one is doing for ordinary homology theories by means of chain complexes. Like chain complexes these chain functors form a category ℭh displaying interesting properties by themselves. The present paper emphasizes more the algebraic aspescts of this category ℭh. Although not every morphism f ∈ ℭh(A*, B*) in the category of chain functors admits a kernel or a cokernel, it turns out that: (1) all cofibrations have a cokernel, (2) all regular fibrations have a kernel, (3) the pushout of a cofibration along a cofibration exists in ℭh, respectively the dual statement for fibrations, (4) there are interesting results about exact sequences involving (co-)fibrations.

On joint spectra of non-commuting hyponormal operators

We show that the left joint spectrum of an arbitrary n-tuple of hyponormal Hilbert space operators can be obtained from the spectral set γ introduced by McIntosh and Pryde. A dual statement for cohyponormal operators is also true. The result is a generalisation of a theorem proved by Pryde and the author for normal operators.

Associative h-spaces and algebras

1. Ganea has shown (Theorem 2·2 of (2)) that h-cogroup structures on a CW space coincide, in the homotopy category, with coalgebra structures defined with respect to the cotriple determined by SΩ. The construction used by Ganea to prove this does not dualize. It seems possible moreover that the dual statement is false. In this note I show, using the results of (4), that, for suitable spaces, there is a surjection from the set of algebra structures defined with respect to the triple determined by ΩS to the set of h-group structures, and I construct a right inverse for this surjection.