Ordering groups constructively
TL;DR: The existence of a linear order on a group that is compatible with the group structure generally requires transfinite methods as mentioned in this paper, and this can be circumvented by concentrating on the cons.
Abstract: The existence of a linear order on a group that is compatible with the group structure generally requires transfinite methods. However, this can be circumvented by concentrating on the cons...
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TL;DR: This paper state and prove a cut-elimination result for inductively generated entailment relations and analyse some of its consequences and describe the existing connections with analogous results in the literature.
Abstract: Entailment relations, introduced by Scott in the early 1970s, provide an abstract generalisation of Gentzen’s multi-conclusion logical inference. Originally applied to the study of multi-valued logics, this notion has then found plenty of applications, ranging from computer science to abstract algebra. In particular, an entailment relation can be regarded as a constructive presentation of a distributive lattice and in this guise it has proven to be a useful tool for the constructive reformulation of several classical theorems in commutative algebra. In this paper, motivated by these concrete applications, we state and prove a cut-elimination result for inductively generated entailment relations. We analyse some of its consequences and describe the existing connections with analogous results in the literature.
16 citations
02 Jul 2019
TL;DR: This paper first gives a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm, and then gives an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.
Abstract: The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn’s lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert’s program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel’s no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.
7 citations
Cites methods from "Ordering groups constructively"
...tion theorems with appropriate syntactic counterparts both sufficient for proofs of elementary statements and provable by elementary means. This method has proved possible in numerous different settings [5,6,21,22,34,44], and in the context of commutative algebra the so-called dynamical method is especially dominant [7,20,45,46].4 In dynamical algebra one deals with a supposed ideal object (such as a maximal ideal) o...
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08 Jul 2020
TL;DR: The characterisation works in the fairly universal setting of a consequence relation enriched with non-deterministic axioms; uniformises many of the known instances of the dynamical method; generalises the proof-theoretic conservation criterion the authors have offered before (with Rinaldi); and links the syntactical with the semantic approach.
Abstract: Dynamical methods were designed to eliminate the ideal objects abstract algebra abounds with. Typically granted by an incarnation of Zorn's Lemma, those ideal objects often serve for proving the semantic conservation of additional non-deterministic sequents, that is, with finite but not necessarily singleton succedents. Eliminating ideal objects dynamically was possible also because (finitary) coherent or geometric logic predominates in that area: the use of a non-deterministic axiom can be captured by a finite branching of the proof tree. Incidentally, a paradigmatic case has widely been neglected in dynamical algebra: Krull's Lemma for prime ideals. Digging deeper just about that case, which we have dealt with only recently (with Yengui), has now brought us to unearth the general phenomenon underlying dynamical algebra: Given a claim of computational nature that usually is proved by the semantic conservation of certain additional non-deterministic axioms, there is a finite labelled tree belonging to a suitable inductively generated class which tree encodes the desired computation. Our characterisation works in the fairly universal setting of a consequence relation enriched with non-deterministic axioms; uniformises many of the known instances of the dynamical method; generalises the proof-theoretic conservation criterion we have offered before (with Rinaldi); and last but not least links the syntactical with the semantic approach: every ideal object used for the customary proof of a concrete claim can be approximated by one of the corresponding tree's branches.
7 citations
TL;DR: In this paper, Camille Jordan et al. describe the conditions générales d'utilisation of the articles of Confluentes Mathematici (http://cml.cedram.org/legal/).
Abstract: © Institut Camille Jordan, 2019, tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http: //cml.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
6 citations
Cites background from "Ordering groups constructively"
...Mithilfe einer alternativen mehrwertigen Schlussrelation, mit welcher die Linearität der korrespondierenden Ordnung durch Hinzunahme von Axiomen ` a,−a für a 6= 0 erzwungen wird, lässt sich eine konstruktive Version des Satzes von Levi erzielen, welche die Inkonsistenz besagter Schlussrelation genau durch die Existenz eines nichttrivialen Torsionselements charakterisiert [46]....
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References
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TL;DR: In this paper, a process for sizing cellulose fibers or cellulose fiber containing materials and a composition for carrying out the process are described, and a method for sizing according to the general formula of R1 is presented.
Abstract: The present invention relates to a process for sizing cellulose fibers or cellulose fiber containing materials and to a composition for carrying out the process. More particularly the invention relates to a process for sizing according to which cellulose fibers or cellulose fiber containing materials in a manner known per se are brought into contact with compounds having the general formula WHEREIN R1 is an organic, hydrophobic group having 8 to 40 carbon atoms and R2 is an alkyl group having 1 to 7 carbon atoms or has the same meaning as R1.
1,915 citations
Book•
17 Nov 2011
1,139 citations
"Ordering groups constructively" refers background in this paper
...For sake of comparison, let us briefly recall one of the well-known classical arguments for Levi’s theorem [28, 40]....
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...We put on record that an ideal element a of ‘ is nothing but the positive cone for a strict linear order < of G that is compatible with the group structure—given a, such an order can be obtained by stipulating [13, 28] a< ab a (1)b 2 a: Indeed, irreflexivity is due to e 62 a, transitivity of < a follows from single-valuedness (s) and multiplicativity (m), and linearity, i....
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...The well-known classical orderability criterion for groups [28, 46, 50] follows from Corollary 3....
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...Then, in Section 3, we study the entailment relation of positive cone of a group, and give a constructive version of the well-known orderability test for groups [28], the contrapositive of which in fact provides the formal Nullstellensatz, i....
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...In particular, our approach does not require us to put forward arguments involving an abstract order conceived as a completed totality—more often than not, the existence of which is intimately linked with a suitable form of the axiom of choice [28, 35]....
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Book•
15 Sep 2011
TL;DR: In this paper, the authors define two definitions of lattices and describe how to describe lattices, including terms, identities, and inequalities, as well as constructions of lattice constructions.
Abstract: Preface.- Introduction.- Glossary of Notation.- I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Basic Concepts.- 4 Terms, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Terms and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- III Congruences.- 1 Congruence Spreading.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- IV Lattice Constructions.- 1 Adding an Element.- 2 Gluing.- 3 Chopped Lattices.- 4 Constructing Lattices with Given Congruence Lattices.- 5 Boolean Triples.- V Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- VI Varieties of Lattices.- 1 Characterizations of Varieties 397.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- VII Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Afterword.- Bibliography.
602 citations
Book•
01 Jan 1989
TL;DR: In this paper, the authors propose a special class of Boolean algebra called superatomic Boolean algebra, which is a subclass of the class of superatomic boolean algebra, and show that it is undecidable to extend the theory of boolean algebra.
Abstract: Special Classes of Boolean Algebras: Superatomic Boolean Algebras (J. Roitman). Projective Boolean Algebras (S. Koppelberg). Countable Boolean Algebras (R.S. Pierce). Measure Algebras (D.H. Fremlin). Logical Questions: Decidable Extensions of the Theory of Boolean Algebras (M. Weese). Undecidable Extensions of the Theory of Boolean Algebras (M. Weese). Recursive Boolean Algebras (J.B. Remmel). Lindenbaum-Tarski Algebras (D. Myers). Boolean-Valued Models (T. Jech). Appendix on Set Theory (J.D. Monk). Chart of Topological Duality. Appendix on General Topology (B. Balcar, P. Simon). References. Bibliography. Index.
462 citations
Book•
07 May 1987TL;DR: A survey of constructive approaches to pure mathematics can be found in this article, where the authors emphasise the viewpoint of Errett Bishop's school, but intuitionism Russian constructivism and recursive analysis are also treated.
Abstract: This is an introduction to, and survey of, the constructive approaches to pure mathematics The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists This account for non-specialists in these and other disciplines
452 citations