From modules to lattices: Insight into the genesis of Dedekind's Dualgruppen

Abstract

When Dedekind introduced the notion of a module, he also defined their divisibility and related arithmetical notions (e.g. the LCM of modules). The introduction of notations for these notions allowed Dedekind to state new theorems, now recognized as the modular laws in lattice theory. Observing the dualism displayed by the theorems, Dedekind pursued his investigations on the matter. This led him, 20 years later, to introduce Dualgruppen, equivalent to lattices Dedekind [Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler. In Dedekind (1930–1932), volume II, 1897, 103–147; Über die von drei Moduln erzeugte Dualgruppe. In Dedekind (1930–1932), volume II, 1900, 236–271]. After a brief exposition of the basic elements of Dualgruppe theory, and with the help of his Nachlass, I show how Dedekind gradually built his theory through layers of computations, often repeated in slight variations and attempted generalizations. I study the tools he devised to help and accompany him in his computations. I highlight the crucial conceptual move that consisted in going from investigating operations between modules, to groups of modules closed under these operations. By using Dedekind's drafts, I aim to highlight the concealed yet essential practices anterior to the published text.

Acknowledgments

I am grateful to the members of the project for their useful remarks. I am also deeply indebted to the members of the ‘Histoire des sciences, histoire du texte’ group of the Laboratoire SPHERE, for many insightful and inspiring discussions. I would finally like to thank two anonymous referees for their comments on this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Emmylou Haffner http://orcid.org/0000-0003-3210-9804

Notes

1 Although a Dualgruppe is formally equivalent to a lattice, the two concepts were developed independently, and are in fact very different in their conception and uses. I will thus use Dedekind's terminology.

2 In fact, Birkhoff's first works on lattices are independent of Dedekind's Dualgruppen. It was Oysten Öre, who was at the time editing Dedekind's Gesammelte Werke, who brought these works to Birkhoff's attention (Birkhoff 1934, 200).

3 See, for example, (Dedekind 1930–1932, III, 468), or (Dedekind 1876–1877a, 102).

4 Edwards (2010).

5 The distinction that I am making here, between the preliminary phases of research and the published work, is slightly different from the traditional distinction between the context of discovery and the context of justification. Indeed, the justification of a result in mathematics is taken to be its proof, which certainly can and does happen during the preliminary phases of research. Rather, it is a distinction between the mathematicians' modes of writing for research (which can be multiple) and for publication (which answer specific disciplinary criteria). It can be linked to the idea that ‘mathematics has a front and a back’ (Hersch 1991).

6 I will use the term ‘module theory’ as an actor's category, that is, as Dedekind himself designates the set of definitions and results he established for modules (and similarly, for example, for ideal theory or Dualgruppe theory), which he considered as fairly clearly defined and circumscribed within his works, from its first introduction in (Dedekind 1871). This should not be taken as meaning that module theory is an established discipline within the mathematical community at the time. As Corry (2004) showed, the meaning of many of the concepts of structural algebra, notably in Dedekind's works, were not fixed at the end of the nineteenth century.

7 In the over 600 folios of manuscripts on module theory, Modulgruppen and Dualgruppen kept in Göttingen archives, less than 200 pages are full of prose (as opposed to mere computations and tables).

8 The term ‘group’, in Dedekind's writings, is used with a relatively large and fluid meaning. Most of the time, Dedekind used the word ‘group’ as he understood it from Galois's works (which he was the first to teach in Germany; Dedekind 1856–1858): a set of elements closed under one or two given operations. The properties of the operations would give defining properties of the ‘group’ in question—properties like associativity were not, for example, included in his general idea of a ‘group’. This is the use made in his drafts on Dualgruppen. In this paper, I will use the term ‘group’ as an actor's category and follow Dedekind's use of the word. Note that outside of Dedekind's works, at the end of the nineteenth century, the term ‘group’ also covered several different meanings or understandings of the concept (see Wussing 1984; Ehrhardt 2012). There was, thus, no ‘group theory’ as we understand it today. Finally, as an anonymous referee pointed out to me, one can also observe, in a draft of Dedekind (1888) reproduced in (Dugac 1976, 296), that Dedekind's chains were first called ‘groups’. This is one of many occurrences in Dedekind's manuscripts of changing terminology as part of the research process—see, below, the changing names of Dualgruppen. One can also mention that in a manuscript from before 1871 (Dedekind around 1870), Dedekind used the word ‘Körper’ to denote a system of points with certain topological properties, which had nothing in common with the algebraic notion of a field introduced in Dedekind (1871).

9 See the ‘Notes on the archival material’ below.

10 In addition, a further step in studying the genesis of Dedekind's Dualgruppen will be a critical genetic edition of the manuscripts (see de Biasi 2011; Grésillon 2016), a work in progress at http://eman-archives.org/Dedekind/ which will allow for a better and more global grasp of the research and writing process.

11 A version of this notation is used in (Dedekind 1897).

12 This notion of Kette, which is also used in (Dedekind 1894, 90), and was largely used in algebra after that, is not related to the concept introduced in (Dedekind 1888) under the same name.

13 Recall that $a^{″}=a+d^{\prime }$ and $a_2={a-d}_1$ (with $d^{\prime }=a^{‴}-b^{‴}-c^{‴}$ and $d_1=a_3+b_3+c_3$ being dual of each other), thus being dual of each other (and so on for $b^{″},c^{″}$ and $b_2,c_2)$. The equalities are equivalent to $(b,c)=(b,a+c)(c+d^{\prime },c)=(b,b-d_1)(b-a,c)$.

14 For the Dualgruppe generated by 3 modules, see (Dedekind 1900, 246–247).

15 It is likely that the $a_3$ indicated on the diagram is a mistake—as supported by the chain given a little further down in the manuscript.

16 I am grateful to Karine Chemla for pointing this out to me.

17 See (Peckhaus 1994).

18 Note that other uses of ‘logic’ are made with reference to Schröder.

19 Dedekind's statement about the non-existence of a logical link between the laws of addition and the Modulgesetz is in his published papers (see above). A notable difference in the published papers is that the absorption law replaces the idempotence property, which is derived from the absorption law.

20 For Dedekind, any set of elements closed under a binary operation was considered a group.

Additional information

Funding

This research was supported by the Deutsche Forschungsgemeinschaft - DFG-Forschungsprojekt ‘Dualität – ein Archetypus mathematischen Denkens’ 279002986.