Editing Modular lattice (section)

==History==

[[File:Free modular lattice with 3 generators (x,y,z).gif|thumb|Free modular lattice generated by three elements {x,y,z}]]
The definition of modularity is due to [[Richard Dedekind]], who published most of the relevant papers after his retirement.
In a paper published in 1894{{citation needed|date=August 2015}} he studied lattices, which he called ''dual groups'' ({{langx|de|Dualgruppen}}) as part of his "algebra of [[Module (mathematics)|modules]]" and observed that ideals satisfy what we now call the modular law. He also observed that for lattices in general, the modular law is equivalent to its dual.

In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility.<ref name="Dedekind.1897">{{Citation | contribution-url=http://digisrv-1.biblio.etc.tu-bs.de:8080/docportal/servlets/MCRFileNodeServlet/DocPortal_derivate_00006737/V.C.1596.pdf | last1=Dedekind | first1=Richard | author1-link=Richard Dedekind | contribution=Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Theiler | title=Festschrift der Herzogl. Technischen Hochschule Carolo-Wilhelmina bei Gelegenheit der 69. Versammlung Deutscher Naturforscher und Ärzte in Braunschweig | publisher=Friedrich Vieweg und Sohn | year=1897 }}</ref>
In a digression he introduced and studied lattices formally in a general context.<ref name="Dedekind.1897"/>{{rp|10–18}} He observed that the lattice of submodules of a module satisfies the modular identity. He called such lattices ''dual groups of module type'' ({{lang|de|Dualgruppen vom Modultypus}}). He also proved that the modular identity and its dual are equivalent.<ref name="Dedekind.1897"/>{{rp|13}}

In the same paper, Dedekind <!---couldn't find that in the source pdf:---observed further that any lattice of ideals of a commutative ring satisfies--->also investigated the following stronger form<ref name="Dedekind.1897"/>{{rp|14}} of the modular identity, which is also self-dual:<ref name="Dedekind.1897"/>{{rp|9}}

: (''x'' ∧ ''b'') ∨ (''a'' ∧ ''b'') = [''x'' ∨ ''a''] ∧ ''b''.<!-- [] and funny choice of variables for consistency with the modular identity -->

He called lattices that satisfy this identity ''dual groups of ideal type'' ({{lang|de|Dualgruppen vom Idealtypus}}).<ref name="Dedekind.1897"/>{{rp|13}} In modern literature, they are more commonly referred to as [[distributive lattice]]s. He gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type.<ref name="Dedekind.1897"/>{{rp|14}}

A paper published by Dedekind in 1900 had lattices as its central topic: He described the free modular lattice generated by three elements, a lattice with 28 elements (see picture).<ref>{{Citation | doi=10.1007/BF01448979 | last1=Dedekind | first1=Richard | title=Über die von drei Moduln erzeugte Dualgruppe | year=1900 | journal=Mathematische Annalen | volume=53 | issue=3 | pages=371–403 | s2cid=122529830 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002257947 }}</ref>