Abstract
This chapter begins with an introductory section which fixes the formal algebraic framework of distributive lattices and of Boolean algebras.
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- 1.
What for Kummer was “the ideal gcd of several numbers” has been replaced in modern language by the corresponding finitely generated ideal. This tour de force, due to Dedekind, was one of the first intrusions of the “actual” infinite in mathematics.
- 2.
Actually the truth values of constructive mathematics do not strictly speaking form a set, but a class. Nevertheless the constructive logical connectives act on those truth values with the same algebraic properties as the \(\wedge \), the \(\vee \) and the \(\rightarrow \) of Heyting algebras. See the discussion on p. 951.
- 3.
Recall that this is a set \(E\) equipped with an order relation \(\le \) for which we have, for all \(x\) and \(y\in E\), \(x\le y\) or \(y\le x\). This does not imply that the equality is decidable.
- 4.
The fact that, when passing to the quotient, we change only the equality relation and not the objects is simpler than the classical approach, and is more consistent with the (Gaussian) tradition and with machine implementation. No doubt the popular success of equivalence classes as objects of the quotient set is largely due to the fortunate fact that in the case of a quotient group \(G/H\), in additive notation for example, we have \((x+H)+(y+H) = (x+y)+H\) where the symbol \(+\) has three different meanings. However, things are less fortunate in the case of quotient rings. For example, \((3+7\mathbb {Z})(2+7\mathbb {Z})\) is contained within, but is not equal to \(6+7\mathbb {Z}\).
- 5.
Actually we need to introduce a restriction to truly obtain a set, in order to have a well-defined procedure to construct concerned ideals. For example we can consider the set of ideals obtained from principal ideals via certain predefined operations, such as countable unions and intersections. This is the same problem as the one indicated in footnote 2.
- 6.
In fact, by direct computation, if \(\varphi (a)=0\), then \(\varphi (\left| {a}\right| )=\left| {\varphi (a)}\right| =0\), and \(\left| {\varphi (x)}\right| =\varphi (\left| {x}\right| )\le \varphi (n\left| {a}\right| )=n \varphi (\left| {a}\right| )=0\), so \(\varphi (x)=0\).
- 7.
More precisely, as \(\preccurlyeq \) is only a preorder, we take for \(\mathbf {T}\) the quotient of \(\mathrm{P}_{\mathrm{fe}}\big (\mathrm{P}_{\mathrm{fe}}(S)\big )\) with respect to the equivalence relation: \(X\preccurlyeq Y\) and \(Y\preccurlyeq X\).
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© 2015 Springer Science+Business Media Dordrecht
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Lombardi, H., Quitté, C. (2015). Distributive Lattices Lattice-Groups. In: Commutative Algebra: Constructive Methods. Algebra and Applications, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9944-7_11
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DOI: https://doi.org/10.1007/978-94-017-9944-7_11
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