109 research outputs found
Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective
We establish, generalizing Di Nola and Lettieri's categorical equivalence, a
Morita-equivalence between the theory of lattice-ordered abelian groups and
that of perfect MV-algebras. Further, after observing that the two theories are
not bi-interpretable in the classical sense, we identify, by considering
appropriate topos-theoretic invariants on their common classifying topos, three
levels of bi-intepretability holding for particular classes of formulas:
irreducible formulas, geometric sentences and imaginaries. Lastly, by
investigating the classifying topos of the theory of perfect MV-algebras, we
obtain various results on its syntax and semantics also in relation to the
cartesian theory of the variety generated by Chang's MV-algebra, including a
concrete representation for the finitely presentable models of the latter
theory as finite products of finitely presentable perfect MV-algebras. Among
the results established on the way, we mention a Morita-equivalence between the
theory of lattice-ordered abelian groups and that of cancellative
lattice-ordered abelian monoids with bottom element.Comment: 54 page
Existence of optimal ultrafilters and the fundamental complexity of simple theories
In the first edition of Classification Theory, the second author
characterized the stable theories in terms of saturation of ultrapowers. Prior
to this theorem, stability had already been defined in terms of counting types,
and the unstable formula theorem was known. A contribution of the ultrapower
characterization was that it involved sorting out the global theory, and
introducing nonforking, seminal for the development of stability theory. Prior
to the present paper, there had been no such characterization of an unstable
class. In the present paper, we first establish the existence of so-called
optimal ultrafilters on Boolean algebras, which are to simple theories as
Keisler's good ultrafilters are to all theories. Then, assuming a supercompact
cardinal, we characterize the simple theories in terms of saturation of
ultrapowers. To do so, we lay the groundwork for analyzing the global structure
of simple theories, in ZFC, via complexity of certain amalgamation patterns.
This brings into focus a fundamental complexity in simple unstable theories
having no real analogue in stability.Comment: The revisions aim to separate the set theoretic and model theoretic
aspects of the paper to make it accessible to readers interested primarily in
one side. We thank the anonymous referee for many thoughtful comment
Eberlein oligomorphic groups
We study the Fourier--Stieltjes algebra of Roelcke precompact,
non-archimedean, Polish groups and give a model-theoretic description of the
Hilbert compactification of these groups. We characterize the family of such
groups whose Fourier--Stieltjes algebra is dense in the algebra of weakly
almost periodic functions: those are exactly the automorphism groups of
-stable, -categorical structures. This analysis is then
extended to all semitopological semigroup compactifications of such a
group: is Hilbert-representable if and only if it is an inverse semigroup.
We also show that every factor of the Hilbert compactification is
Hilbert-representable.Comment: 23 page
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