79 research outputs found
Existentially closed De Morgan algebras
We show that the theory of De Morgan algebras has a model completion and axiomatise it. Then we prove that it is ℵ0-categorical and describe definable and algebraic closures in that theory. We also obtain similar results for Boole–De Morgan algebras
Bohrification
New foundations for quantum logic and quantum spaces are constructed by
merging algebraic quantum theory and topos theory. Interpreting Bohr's
"doctrine of classical concepts" mathematically, given a quantum theory
described by a noncommutative C*-algebra A, we construct a topos T(A), which
contains the "Bohrification" B of A as an internal commutative C*-algebra. Then
B has a spectrum, a locale internal to T(A), the external description S(A) of
which we interpret as the "Bohrified" phase space of the physical system. As in
classical physics, the open subsets of S(A) correspond to (atomic)
propositions, so that the "Bohrified" quantum logic of A is given by the
Heyting algebra structure of S(A). The key difference between this logic and
its classical counterpart is that the former does not satisfy the law of the
excluded middle, and hence is intuitionistic. When A contains sufficiently many
projections (e.g. when A is a von Neumann algebra, or, more generally, a
Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be
compared with the traditional quantum logic, i.e. the orthomodular lattice of
projections in A. This time, the main difference is that the former is
distributive (even when A is noncommutative), while the latter is not.
This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in
"Deep Beauty" (ed. H. Halvorson
Bohrification of operator algebras and quantum logic
Following Birkhoff and von Neumann, quantum logic has traditionally been
based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A.
Unfortunately, the logical interpretation of these lattices is impaired by
their nondistributivity and by various other problems. We show that a possible
resolution of these difficulties, suggested by the ideas of Bohr, emerges if
instead of single projections one considers elementary propositions to be
families of projections indexed by a partially ordered set C(A) of appropriate
commutative subalgebras of A. In fact, to achieve both maximal generality and
ease of use within topos theory, we assume that A is a so-called Rickart
C*-algebra and that C(A) consists of all unital commutative Rickart
C*-subalgebras of A. Such families of projections form a Heyting algebra in a
natural way, so that the associated propositional logic is intuitionistic:
distributivity is recovered at the expense of the law of the excluded middle.
Subsequently, generalizing an earlier computation for n-by-n matrices, we
prove that the Heyting algebra thus associated to A arises as a basis for the
internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the
"Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of
functors from C(A) to the category of sets. We explain the relationship of this
construction to partial Boolean algebras and Bruns-Lakser completions. Finally,
we establish a connection between probability measure on the lattice of
projections on a Hilbert space H and probability valuations on the internal
Gelfand spectrum of A for A = B(H).Comment: 31 page
Semigroup-valued metric spaces
The structural Ramsey theory is a field on the boundary of combinatorics and
model theory with deep connections to topological dynamics. Most of the known
Ramsey classes in finite binary symmetric relational language can be shown to
be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's
-metric spaces, Conant's generalised metric spaces, Braunfeld's
-ultrametric spaces or Cherlin's metrically homogeneous graphs). In
this thesis we explore the limits of the shortest path completion. We offer a
unifying framework --- semigroup-valued metric spaces --- for all the
aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the
extension property for partial automorphisms). Our results can be seen as
evidence for the importance of studying the completion problem for amalgamation
classes and have some further applications (such as the stationary independence
relation).
As a corollary of our general theorems, we reprove results of Hubi\v{c}ka and
Ne\v{s}et\v{r}il on Sauer's -metric spaces, results of Hub\v{c}ka,
Ne\v{s}et\v{r}il and the author on Conant's generalised metric spaces,
Braunfeld's results on -ultrametric spaces and the results of Aranda
et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We
also solve several open problems such as EPPA for -ultrametric spaces,
-metric spaces or Conant's generalised metric spaces.
Our framework seems to be universal enough that we conjecture that every
primitive strong amalgamation class of complete edge-labelled graphs with
finitely many labels is in fact a class of semigroup-valued metric spaces.Comment: Master thesis, defended in June 201
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