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A logical approach to abstract algebra
 Math. Structures Comput. Sci
"... Abstract. Recent work in constructive mathematics show that Hilbert’s program works for a large part of abstract algebra. Using in an essential way the ideas contained in the classical arguments, we can transform a large number of abstract non effective proofs of “concrete ” statements into elementa ..."
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Abstract. Recent work in constructive mathematics show that Hilbert’s program works for a large part of abstract algebra. Using in an essential way the ideas contained in the classical arguments, we can transform a large number of abstract non effective proofs of “concrete ” statements into elementary proofs. Surprisingly the arguments we get are not only elementary but also mathematically clearer and not necessarily longer. We present an example where the simplification was significant enough to suggest an improved version of a classical theorem.
On Seminormality
 J. Algebra
"... We give an elementary and essentially selfcontained proof 1 that a reduced ring R is seminormal if and only if the canonical map Pic R → Pic R[X] is an isomorphism, a theorem due to Swan [15], generalizing some previous results of Traverso [16]. By a simple modification of this argument, we obtain ..."
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We give an elementary and essentially selfcontained proof 1 that a reduced ring R is seminormal if and only if the canonical map Pic R → Pic R[X] is an isomorphism, a theorem due to Swan [15], generalizing some previous results of Traverso [16]. By a simple modification of this argument, we obtain a constructive proof, and hence an algorithm [12], associated to a classical proof which is not so easy otherwise to access, since it requires a journey through [15, 16, 1] or, in the domain case, through [14, 13, 2, 6, 7]. We recall [15] that R is seminormal if and only if if b 2 = c 3 then there exists a ∈ R such that b = a 3 and c = a 2. This is a remarkably simple (and technically firstorder) condition. Similarly, as we will show in this note, the statement that the canonical map Pic R → Pic R[X] is an isomorphism can also be formulated in an elementary way. Swan’s original definition includes that R is reduced, but, as noticed by Costa [4], reduceness follows from seminormality: if d 2 = 0 then d 2 = d 3 = 0 and so there exists a ∈ R such that d = a 2 = a 3. We have then d = aa 2 = ad and so d = a(ad) = d 2 = 0. Section 7 of Chapter VIII of [9] surveys the work on commutative seminormal ring up to day. 1 General Lemmas
Relationships between constructive, predicative, and classical systems of analysis
 In Hendricks et al
"... Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded ..."
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Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative " de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives to classical, settheoretically based mathematics until the 1960s. Now wehave a massive amount of information, to which this lecture will constitute an introduction, about what can be done by what means, and about the theoretical interrelationships between various formal systems for constructive, predicative and classical analysis. In this nal lecture I will be sketching some redevelopments of classical analysis on both constructive and predicative grounds, with an emphasis on modern approaches. In the case of constructivity, Ihave very little to say about Brouwerian intuitionism, which has been discussed extensively in other lectures at this conference, and concentrate instead on the approach since 1967 of Errett Bishop and his school. In the case of predicativity, I concentrate on developmentsalso since the 1960swhich take up where Weyl's work left o, as described in my second lecture. In both cases, I rst look at these redevelopments from a more informal, mathematical, point This is the last of my three lectures for the conference, Proof Theory: History and
Constructive Completions of Ordered Sets, Groups and Fields
, 2003
"... In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, so called pseudoorder. ..."
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In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, so called pseudoorder.
A Note on Irreducible Decomposition of Algebraic Sets and Automatic Theorem Proving
, 1996
"... In this paper, we present some new results concerning the dimension of the irreducible components of an algebraic set. We also point out that our results are very useful in Computer Aided Geometric Design (CAGD) and Visualization, in particular when the algebraic set is onedimensional. In the secon ..."
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In this paper, we present some new results concerning the dimension of the irreducible components of an algebraic set. We also point out that our results are very useful in Computer Aided Geometric Design (CAGD) and Visualization, in particular when the algebraic set is onedimensional. In the second part of the paper, we report our experiments on combining results in Automatic Theorem Proving with the computer algebra system CASA  a computer algebra system for Algebraic Geometry. Keywords: Irreducible decomposition, Automatic Theorem Proving . 1 Introduction Irreducible decomposition is widely used in Algebraic Geometry, Computer Aided Geometric Design, etc. In fact, it has been used as the first and requisite step in many constructive computations such as parameterization, genus computation, projection of an algebraic set onto a hypersurface. It is a wellknown result that we can algorithmically decompose an algebraic set (in K n , where K is a field) into a union of finitely man...
Constructive Logic in Algebra
, 2008
"... This document contains two examples of the use of distributive lattices as spaces in commutative algebra. The first example is a simple proof of Forster’s Theorem about the number of generators over a ring of finite Krull dimension. The second example is the beginning of the theory of Prüfer Domain, ..."
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This document contains two examples of the use of distributive lattices as spaces in commutative algebra. The first example is a simple proof of Forster’s Theorem about the number of generators over a ring of finite Krull dimension. The second example is the beginning of the theory of Prüfer Domain, which has to be thought of as a non Noetherian version of the theory of Dedekind
Constructive logic and type theory ∗
, 2004
"... Constructive logic is based on the leading principle that (I) proofs are programs. In constructive type theory it has later been joined with the further principle that (II) propositions are data types. The latter is also called the propositionsastypes principle. The meaning of these principles is ..."
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Constructive logic is based on the leading principle that (I) proofs are programs. In constructive type theory it has later been joined with the further principle that (II) propositions are data types. The latter is also called the propositionsastypes principle. The meaning of these principles is on first sight far from obvious. The purpose of these lectures is to explain these and their consequences and applications in computer science. A particular aim is to provide background for the study of MartinLöf type theory, or similar theories of dependent types, and for using proofchecking systems based on such theories. Proofs carried out within constructive logic may be considered as programs in a functional language, closely related to e.g. ML or Haskell. The importance of this is the possibility to extract from an existence proof (that, e.g., there are arbitrarily large prime numbers) a program that finds or constructs the purported object, and further obtain a verification that the program terminates (finds some number) and is correct (finds only sufficiently large prime numbers).
Author manuscript, published in "CALCULEMUS 2007, Hagenberg: Austria (2007)" DOI: 10.1007/9783540730866_4 Towards Constructive Homological Algebra in Type Theory
, 2009
"... Abstract. This paper reports on ongoing work on the project of representing the Kenzo system [15] in type theory [11]. ..."
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Abstract. This paper reports on ongoing work on the project of representing the Kenzo system [15] in type theory [11].