176 research outputs found
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
Finitely generated free Heyting algebras via Birkhoff duality and coalgebra
Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and
thus the free algebras can be obtained by a direct limit process. Dually, the
final coalgebras can be obtained by an inverse limit process. In order to
explore the limits of this method we look at Heyting algebras which have mixed
rank 0-1 axiomatizations. We will see that Heyting algebras are special in that
they are almost rank 1 axiomatized and can be handled by a slight variant of
the rank 1 coalgebraic methods
A non-standard analysis of a cultural icon: The case of Paul Halmos
We examine Paul Halmos' comments on category theory, Dedekind cuts, devil
worship, logic, and Robinson's infinitesimals. Halmos' scepticism about
category theory derives from his philosophical position of naive set-theoretic
realism. In the words of an MAA biography, Halmos thought that mathematics is
"certainty" and "architecture" yet 20th century logic teaches us is that
mathematics is full of uncertainty or more precisely incompleteness. If the
term architecture meant to imply that mathematics is one great solid castle,
then modern logic tends to teach us the opposite lession, namely that the
castle is floating in midair. Halmos' realism tends to color his judgment of
purely scientific aspects of logic and the way it is practiced and applied. He
often expressed distaste for nonstandard models, and made a sustained effort to
eliminate first-order logic, the logicians' concept of interpretation, and the
syntactic vs semantic distinction. He felt that these were vague, and sought to
replace them all by his polyadic algebra. Halmos claimed that Robinson's
framework is "unnecessary" but Henson and Keisler argue that Robinson's
framework allows one to dig deeper into set-theoretic resources than is common
in Archimedean mathematics. This can potentially prove theorems not accessible
by standard methods, undermining Halmos' criticisms.
Keywords: Archimedean axiom; bridge between discrete and continuous
mathematics; hyperreals; incomparable quantities; indispensability; infinity;
mathematical realism; Robinson.Comment: 15 pages, to appear in Logica Universali
Equivariant quantum cohomology and Yang-Baxter algebras
There are two intriguing statements regarding the quantum cohomology of
partial flag varieties. The first one relates quantum cohomology to the
affinisation of Lie algebras and the homology of the affine Grassmannian, the
second one connects it with the geometry of quiver varieties. The connection
with the affine Grassmannian was first discussed in unpublished work of
Peterson and subsequently proved by Lam and Shimozono. The second development
is based on recent works of Nekrasov, Shatashvili and of Maulik, Okounkov
relating the quantum cohomology of Nakajima varieties with integrable systems
and quantum groups. In this article we explore for the simplest case, the
Grassmannian, the relation between the two approaches. We extend the definition
of the integrable systems called vicious and osculating walkers to the
equivariant setting and show that these models have simple expressions in a
particular representation of the affine nil-Hecke ring. We compare this
representation with the one introduced by Kostant and Kumar and later used by
Peterson in his approach to Schubert calculus. We reveal an underlying quantum
group structure in terms of Yang-Baxter algebras and relate them to Schur-Weyl
duality. We also derive new combinatorial results for equivariant Gromov-Witten
invariants such as an explicit determinant formula.Comment: 60 pages, 11 figures; v2: statement about Schur-Weyl duality added
and introduction slightly rewritten, see last paragraph on page 2 and first
paragraph on page 3 as well as Theorem 1.3 in the introductio
HalmazelmĂ©let; PartĂciĂł kalkulus, VĂ©gtelen gráfok elmĂ©lete = Set Theory; Partition Calculus , Theory of Infinite Graphs
ElĹ‘zetes tervĂĽnknek megfelelĹ‘en a halmazelmĂ©let alábbi terĂĽletein vĂ©geztĂĽnk kutatást Ă©s Ă©rtĂĽnk el számos eredmĂ©nyt: I. Kombinatorika II. A valĂłsak számsosságinvariánsai Ă©s ideálelmĂ©let III. HalmazelmĂ©leti topolĂłgia Ezek mellett Sági Gábor kiterjedt kutatást vĂ©gzett a modellelmĂ©let terĂĽletĂ©n , amely eredmĂ©nyek kapcsolĂłdnak a kombinatorikához is. EredmĂ©nyeinket 38 közlemĂ©nyben publikáltuk, amelyek majdnem mind az adott terĂĽlet vezetĹ‘ nemzetközi lapjaiban jelentel meg (5 cikket csak benyĂşjtottunk). Számos nemzetközi konferencián is rĂ©sztvettĂĽnk, Ă©s hárman közűlĂĽnk (Juhász, Sádi, Soukup) plenáris/meghĂvott elĹ‘adĂłk voltak számos alkalommal. | Following our research plan, we have mainly done research -- and established a number of significant results -- in several areas of set theory: I. Combinatorics II. Cardinal invariants of the continuum and ideal theory III. Set-theoretic topology In addition to these, G. Sági has done extended research in model theory that had ramifications to combinatorics. We presented our results in 38 publications, almost all of which appeared or will appear in the leading international journals of these fields (5 of these papers have been submitted but not accepted as yet). We also participated at a number of international conferences, three of us (Juhász, Sági, Soukup) as plenary and/or invited speakers at many of these
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
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