146 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
Unital hyperarchimedean vector lattices
We prove that the category of unital hyperarchimedean vector lattices is
equivalent to the category of Boolean algebras. The key result needed to
establish the equivalence is that, via the Yosida representation, such a vector
lattice is naturally isomorphic to the vector lattice of all locally constant
real-valued continuous functions on a Boolean (=compact Hausdorff totally
disconnected) space. We give two applications of our main result.Comment: 15 pages. Submitted pape
Probabilistic logics based on Riesz spaces
We introduce a novel real-valued endogenous logic for expressing properties
of probabilistic transition systems called Riesz modal logic. The design of the
syntax and semantics of this logic is directly inspired by the theory of Riesz
spaces, a mature field of mathematics at the intersection of universal algebra
and functional analysis. By using powerful results from this theory, we develop
the duality theory of the Riesz modal logic in the form of an
algebra-to-coalgebra correspondence. This has a number of consequences
including: a sound and complete axiomatization, the proof that the logic
characterizes probabilistic bisimulation and other convenient results such as
completion theorems. This work is intended to be the basis for subsequent
research on extensions of Riesz modal logic with fixed-point operators
Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras
The assignment (nonstable K_0-theory), that to a ring R associates the monoid
V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices
with only finitely nonzero entries over R, extends naturally to a functor. We
prove the following lifting properties of that functor: (1) There is no functor
F, from simplicial monoids with order-unit with normalized positive
homomorphisms to exchange rings, such that VF is equivalent to the identity.
(2) There is no functor F, from simplicial monoids with order-unit with
normalized positive embeddings to C*-algebras of real rank 0 (resp., von
Neumann regular rings), such that VF is equivalent to the identity. (3) There
is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be
lifted, with respect to the functor V, by exchange rings and by C*-algebras of
real rank 1, but not by semiprimitive exchange rings, thus neither by regular
rings nor by C*-algebras of real rank 0. By using categorical tools from an
earlier paper (larders, lifters, CLL), we deduce that there exists a unital
exchange ring of cardinality aleph three (resp., an aleph three-separable
unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence
2, such that V(R) is the positive cone of a dimension group and V(R) is not
isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0
or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea
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