26,567 research outputs found
Towards a Model Theory for Transseries
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
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
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 of an ordered monoid
\tM is replaced by R: = L\times \tM, where is a given indexing semiring
(not necessarily with 0). In this case we say layered by . When is
trivial, i.e, , is the usual bipotent max-plus algebra. When
we recover the "standard" supertropical structure with its
"ghost" layer. When L = \NN we can describe multiple roots of polynomials
via a "layering function" . Likewise, one can define the layering
componentwise; vectors 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
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
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
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 -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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