Compatible Tight Riesz Orders

N. R. Reilly has obtained an algebraic characterization of the compatible tight Riesz orders that can be supported by certain partially ordered groups [13; 14]. The purpose of this paper is to give a “geometric“ characterization by the use of ordered permutation groups. Our restrictions on the partially ordered groups will likewise be geometric rather than algebraic. Davis and Bolz [3] have done some work on groups of all order-preserving permutations of a totally ordered field; from our more general theorems, we will be able to recapture their results.

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From Tarski to Descartes: Formalization of the Arithmetization of Euclidean Geometry

10.29007/k47p ◽

2018 ◽

Cited By ~ 1

Author(s):

Pierre Boutry ◽

Gabriel Braun ◽

Julien Narboux

Keyword(s):

Congruence Relation ◽

Euclidean Geometry ◽

Formal Proof ◽

Automated Deduction ◽

Algebraic Characterization ◽

Proof Assistant ◽

Ordered Field ◽

The Coq Proof Assistant ◽

Coq Proof Assistant

This paper describes the formalization of the arithmetization of Euclidean geometry in the Coq proof assistant.As a basis for this work, Tarski’s system of geometry was chosen for its well-known metamathematical properties.This work completes our formalization of the two-dimensional results contained in part one of Metamathematische Methoden in der Geometrie.We define the arithmetic operations geometrically and prove that they verify the properties of an ordered field.Then, we introduce cartesian coordinates, and provide an algebraic characterization of the main geometric predicates.In order to prove the characterization of the segment congruence relation, we provide a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems.The arithmetization of geometry justifies the use the algebraic automated deduction methods in geometry.We give an example of the use this formalization by deriving from Tarski's system of geometry a formal proof of theorems of nine points using Gröbner basis.

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The Algebraic Characterization of Geometric 4-Manifolds

10.1017/cbo9780511526350 ◽

1994 ◽

Cited By ~ 14

Author(s):

J. A. Hillman

Keyword(s):

Algebraic Characterization

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ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

Nagoya Mathematical Journal ◽

10.1017/nmj.2021.6 ◽

2021 ◽

pp. 1-40

Author(s):

NICK GILL ◽

BIANCA LODÀ ◽

PABLO SPIGA

Keyword(s):

Permutation Group ◽

Model Theory ◽

Independent Set ◽

Maximum Size ◽

Proper Subset ◽

Permutation Groups ◽

Finite Permutation Group ◽

Pointwise Stabilizer ◽

Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

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Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

Mathematics ◽

10.3390/math9070728 ◽

2021 ◽

Vol 9 (7) ◽

pp. 728

Author(s):

Yasunori Maekawa ◽

Yoshihiro Ueda

Keyword(s):

Hyperbolic System ◽

Dissipative Structure ◽

Dissipative Structures ◽

Algebraic Characterization ◽

Symmetric Hyperbolic System ◽

Full Generality ◽

Order Linear ◽

First Order

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.

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Intrinsic metric preserving maps on partially ordered groups

Algebra Universalis ◽

10.1007/bf01192713 ◽

1996 ◽

Vol 36 (1) ◽

pp. 135-140 ◽

Cited By ~ 5

Author(s):

M. Jasem

Keyword(s):

Intrinsic Metric ◽

Partially Ordered ◽

Ordered Groups

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An Axiomatic Characterization of a Class of Petri Nets

Fundamenta Informaticae ◽

10.3233/fi-1991-14405 ◽

1991 ◽

Vol 14 (4) ◽

pp. 477-491

Author(s):

Waldemar Korczynski

Keyword(s):

Petri Nets ◽

Petri Net ◽

Axiomatic Characterization ◽

Algebraic Characterization

In this paper an algebraic characterization of a class of Petri nets is given. The nets are characterized by a kind of algebras, which can be considered as a generalization of the concept of the case graph of a (marked) Petri net.

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