| 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. |
