18 research outputs found
Stone duality above dimension zero: Axiomatising the algebraic theory of C(X)
It has been known since the work of Duskin and Pelletier four decades ago
that KH^op, the category opposite to compact Hausdorff spaces and continuous
maps, is monadic over the category of sets. It follows that KH^op is equivalent
to a possibly infinitary variety of algebras V in the sense of Slominski and
Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of V can
be generated using a finite number of finitary operations, together with a
single operation of countably infinite arity. In 1983, Banaschewski and Rosicky
independently proved a conjecture of Bankston, establishing a strong negative
result on the axiomatisability of KH^op. In particular, V is not a finitary
variety--Isbell's result is best possible. The problem of axiomatising V by
equations has remained open. Using the theory of Chang's MV-algebras as a key
tool, along with Isbell's fundamental insight on the semantic nature of the
infinitary operation, we provide a finite axiomatisation of V.Comment: 26 pages. Presentation improve
Beth definability and the Stone-Weierstrass Theorem
The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result
of functional analysis with far-reaching consequences. We show that this
theorem is a consequence of the Beth definability property of a certain
infinitary equational logic, stating that every implicit definition can be made
explicit.Comment: 20 pages. v2: minor changes, added a "Conclusion" sectio
Sums of two dimensional spectral triples
We study countable sums of two dimensional modules for the continuous complex
functions on a compact metric space and show that it is possible to construct a
spectral triple which gives the original metric back. This spectral triple will
be finitely summable for any positive parameter. We also construct a sum of two
dimensional modules which reflects some aspects of the topological dimensions
of the compact metric space, but this will only give the metric back
approximately. We make an explicit computation of the last module for the unit
interval. The metric is recovered exactly, the Dixmier trace induces a multiple
of the Lebesgue integral and the number N(K) of eigenvalues bounded by K
behaves, such that N(K)/K is bounded, but without limit for K growing.Comment: 27 page