100 research outputs found
A topological theory of (T,V)-categories
Lawvere's notion of completeness for quantale-enriched categories has been extended to the theory of lax algebras under the name of L-completeness. In this work we introduce the corresponding morphism concept and examine its properties. We explore some important relativized topological concepts like separation, density, compactness and compactification with respect to L-complete morphisms. We show that separated L-complete morphisms belong to a factorization system. Moreover, we investigate relativized topological concepts with respect to maps that preserve L-closure which is the natural symmetrized closure for lax algebras. We provide concrete characterizations of Zariski closure and Zariski compactness for approach spaces
Prime Ideal Theorems and systems of finite character
summary:\font\jeden=rsfs7 \font\dva=rsfs10 We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if \text{\jeden S} is a system of finite character then so is the system of all collections of finite subsets of \bigcup \text{\jeden S} meeting a common member of \text{\jeden S}), the Finite Cutset Lemma (a finitary version of the Teichm"uller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
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On the Logic of Belief and Propositional Quantification
We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss
Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry
This paper surveys results related to well-known works of B. Plotkin and V.
Remeslennikov on the edge of algebra, logic and geometry. We start from a brief
review of the paper and motivations. The first sections deal with model theory.
In Section 2.1 we describe the geometric equivalence, the elementary
equivalence, and the isotypicity of algebras. We look at these notions from the
positions of universal algebraic geometry and make emphasis on the cases of the
first order rigidity. In this setting Plotkin's problem on the structure of
automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of
categories is pretty natural and important. Section 2.2 is dedicated to
particular cases of Plotkin's problem. Section 2.3 is devoted to Plotkin's
problem for automorphisms of the group of polynomial symplectomorphisms. This
setting has applications to mathematical physics through the use of model
theory (non-standard analysis) in the studying of homomorphisms between groups
of symplectomorphisms and automorphisms of the Weyl algebra. The last two
sections deal with algorithmic problems for noncommutative and commutative
algebraic geometry. Section 3.1 is devoted to the Gr\"obner basis in
non-commutative situation. Despite the existence of an algorithm for checking
equalities, the zero divisors and nilpotency problems are algorithmically
unsolvable. Section 3.2 is connected with the problem of embedding of algebraic
varieties; a sketch of the proof of its algorithmic undecidability over a field
of characteristic zero is given.Comment: In this review we partially used results of arXiv:1512.06533,
arXiv:math/0512273, arXiv:1812.01883 and arXiv:1606.01566 and put them in a
new contex
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
Rank rigidity for CAT(0) cube complexes
We prove that any group acting essentially without a fixed point at infinity
on an irreducible finite-dimensional CAT(0) cube complex contains a rank one
isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube
complexes. We derive a number of other consequences for CAT(0) cube complexes,
including a purely geometric proof of the Tits Alternative, an existence result
for regular elements in (possibly non-uniform) lattices acting on cube
complexes, and a characterization of products of trees in terms of bounded
cohomology.Comment: 39 pages, 4 figures. Revised version according to referee repor
Interpretations of ZF
In this paper, we unify the study of classical and non-classical
algebra-valued models of set theory, by studying variations of the
interpretation functions for identity and set-membership. Although, these
variations coincide with the standard interpretation in Boolean-valued
constructions, nonetheless they extend the scope of validity of ZF to new
algebra-valued models
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