514,667 research outputs found
Chain Decomposition Theorems for Ordered Sets (and Other Musings)
A brief introduction to the theory of ordered sets and lattice theory is
given. To illustrate proof techniques in the theory of ordered sets, a
generalization of a conjecture of Daykin and Daykin, concerning the structure
of posets that can be partitioned into chains in a ``strong'' way, is proved.
The result is motivated by a conjecture of Graham's concerning probability
correlation inequalities for linear extensions of finite posets
Generalized Integer Partitions, Tilings of Zonotopes and Lattices
In this paper, we study two kinds of combinatorial objects, generalized
integer partitions and tilings of two dimensional zonotopes, using dynamical
systems and order theory. We show that the sets of partitions ordered with a
simple dynamics, have the distributive lattice structure. Likewise, we show
that the set of tilings of zonotopes, ordered with a simple and classical
dynamics, is the disjoint union of distributive lattices which we describe. We
also discuss the special case of linear integer partitions, for which other
dynamical systems exist. These results give a better understanding of the
behaviour of tilings of zonotopes with flips and dynamical systems involving
partitions.Comment: See http://www.liafa.jussieu.fr/~latapy
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
Scheme theoretic tropicalization
In this paper, we introduce ordered blueprints and ordered blue schemes,
which serve as a common language for the different approaches to
tropicalizations and which enhances tropical varieties with a schematic
structure. As an abstract concept, we consider a tropicalization as a moduli
problem about extensions of a given valuation between ordered
blueprints and . If is idempotent, then we show that a
generalization of the Giansiracusa bend relation leads to a representing object
for the tropicalization, and that it has yet another interpretation in terms of
a base change along . We call such a representing object a scheme theoretic
tropicalization.
This theory recovers and improves other approaches to tropicalizations as we
explain with care in the second part of this text.
The Berkovich analytification and the Kajiwara-Payne tropicalization appear
as rational point sets of a scheme theoretic tropicalization. The same holds
true for its generalization by Foster and Ranganathan to higher rank
valuations.
The scheme theoretic Giansiracusa tropicalization can be recovered from the
scheme theoretic tropicalizations in our sense. We obtain an improvement due to
the resulting blueprint structure, which is sufficient to remember the
Maclagan-Rinc\'on weights.
The Macpherson analytification has an interpretation in terms of a scheme
theoretic tropicalization, and we give an alternative approach to Macpherson's
construction of tropicalizations.
The Thuillier analytification and the Ulirsch tropicalization are rational
point sets of a scheme theoretic tropicalization. Our approach yields a
generalization to any, possibly nontrivial, valuation with
idempotent and enhances the tropicalization with a schematic structure.Comment: 66 pages; for information about the changes in this version of the
paper, please cf. the paragraph "Differences to previous versions" in the
introductio
Borel and countably determined reducibility in nonstandard domain
We consider reducibility of equivalence relations (ERs, for brevity), in a
nonstandard domain, in terms of the Borel reducibility and the countably
determined (CD, for brevity) reducibility. This reveals phenomena partially
analogous to those discovered in descriptive set theory. The Borel reducibility
structure of Borel sets and (partially) CD reducibility structure of CD sets in
*N is described. We prove that all CD ERs with countable equivalence classes
are CD-smooth, but not all are B-smooth, for instance, the ER of having finite
difference on *N. Similarly to the Silver dichotomy theorem in Polish spaces,
any CD ER on *N either has at most continuum-many classes or there is an
infinite internal set of pairwise inequivalent elements. Our study of monadic
ERs on *N, i.e., those of the form x E y iff |x-y| belongs to a given additive
Borel cut in *N, shows that these ERs split in two linearly families,
associated with countably cofinal and countably coinitial cuts, each of which
is linearly ordered by Borel reducibility. The relationship between monadic ERs
and the ER of finite symmetric difference on hyperfinite subsets of *N is
studied.Comment: 34 page
Models of Type Theory Based on Moore Paths
This paper introduces a new family of models of intensional Martin-L\"of type
theory. We use constructive ordered algebra in toposes. Identity types in the
models are given by a notion of Moore path. By considering a particular gros
topos, we show that there is such a model that is non-truncated, i.e. contains
non-trivial structure at all dimensions. In other words, in this model a type
in a nested sequence of identity types can contain more than one element, no
matter how great the degree of nesting. Although inspired by existing
non-truncated models of type theory based on simplicial and cubical sets, the
notion of model presented here is notable for avoiding any form of Kan filling
condition in the semantics of types.Comment: This is a revised and expanded version of a paper with the same name
that appeared in the proceedings of the 2nd International Conference on
Formal Structures for Computation and Deduction (FSCD 2017
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