26,567 research outputs found

    Towards a Model Theory for Transseries

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    The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field, and report on our efforts to understand its first-order theory.Comment: Notre Dame J. Form. Log., to appear; 33 p

    Towards "dynamic domains": totally continuous cocomplete Q-categories

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    It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or more generally, a quantaloid. In fact, we are lead to consider cocomplete quantaloid-enriched categories as fundamental mathematical structure for a dynamic logic common to both computer science and physics. Here we explain the theory of totally continuous cocomplete categories as generalization of the well-known theory of totally continuous suplattices. That is to say, we undertake some first steps towards a theory of "dynamic domains''.Comment: 29 pages; contains a more elaborate introduction, corrects some typos, and has a sexier title than the previously posted version, but the mathematics are essentially the sam

    Algebraic structures of tropical mathematics

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    Tropical mathematics often is defined over an ordered cancellative monoid \tM, usually taken to be (\RR, +) or (\QQ, +). Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra AA of an ordered monoid \tM is replaced by R: = L\times \tM, where LL is a given indexing semiring (not necessarily with 0). In this case we say RR layered by LL. When LL is trivial, i.e, L={1}L=\{1\}, RR is the usual bipotent max-plus algebra. When L={1,∞}L=\{1,\infty\} we recover the "standard" supertropical structure with its "ghost" layer. When L = \NN we can describe multiple roots of polynomials via a "layering function" s:R→Ls: R \to L. Likewise, one can define the layering s:R(n)→L(n)s: R^{(n)} \to L^{(n)} componentwise; vectors v1,…,vmv_1, \dots, v_m are called tropically dependent if each component of some nontrivial linear combination \sum \a_i v_i is a ghost, for "tangible" \a_i \in R. Then an n×nn\times n matrix has tropically dependent rows iff its permanent is a ghost. We explain how supertropical algebras, and more generally layered algebras, provide a robust algebraic foundation for tropical linear algebra, in which many classical tools are available. In the process, we provide some new results concerning the rank of d-independent sets (such as the fact that they are semi-additive),put them in the context of supertropical bilinear forms, and lay the matrix theory in the framework of identities of semirings.Comment: 19 page

    Proof Theory and Ordered Groups

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    Ordering theorems, characterizing when partial orders of a group extend to total orders, are used to generate hypersequent calculi for varieties of lattice-ordered groups (l-groups). These calculi are then used to provide new proofs of theorems arising in the theory of ordered groups. More precisely: an analytic calculus for abelian l-groups is generated using an ordering theorem for abelian groups; a calculus is generated for l-groups and new decidability proofs are obtained for the equational theory of this variety and extending finite subsets of free groups to right orders; and a calculus for representable l-groups is generated and a new proof is obtained that free groups are orderable

    Notes on divisible MV-algebras

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    In these notes we study the class of divisible MV-algebras inside the algebraic hierarchy of MV-algebras with product. We connect divisible MV-algebras with Q\mathbb Q-vector lattices, we present the divisible hull as a categorical adjunction and we prove a duality between finitely presented algebras and rational polyhedra
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