100 research outputs found

    A topological theory of (T,V)-categories

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    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

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    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

    Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry

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    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

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    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

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    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

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    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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