History
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 he studied lattices, which he called dual groups (German: Dualgruppen) as part of his "algebra of 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. In a digression he introduced and studied lattices formally in a general context. He observed that the lattice of submodules of a module satisfies the modular identity. He called such lattices dual groups of module type (German: Dualgruppen vom Modultypus). He also proved that the modular identity and its dual are equivalent.
In the same paper, Dedekind observed further that any lattice of ideals of a commutative ring satisfies the following stronger form of the modular identity, which is also self-dual:
- (x ∧ b) ∨ (a ∧ b) = ∧ b.
He called lattices that satisfy this identity dual groups of ideal type (German: Dualgruppen vom Idealtypus). In modern literature, they are more commonly referred to as distributive lattices. He gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type.
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.
Read more about this topic: Modular Lattice
Famous quotes containing the word history:
“The history of all Magazines shows plainly that those which have attained celebrity were indebted for it to articles similar in natureto Berenicealthough, I grant you, far superior in style and execution. I say similar in nature. You ask me in what does this nature consist? In the ludicrous heightened into the grotesque: the fearful coloured into the horrible: the witty exaggerated into the burlesque: the singular wrought out into the strange and mystical.”
—Edgar Allan Poe (18091849)
“No one is ahead of his time, it is only that the particular variety of creating his time is the one that his contemporaries who are also creating their own time refuse to accept.... For a very long time everybody refuses and then almost without a pause almost everybody accepts. In the history of the refused in the arts and literature the rapidity of the change is always startling.”
—Gertrude Stein (18741946)
“It is my conviction that women are the natural orators of the race.”
—Eliza Archard Connor, U.S. suffragist. As quoted in History of Woman Suffrage, vol. 4, ch. 9, by Susan B. Anthony and Ida Husted Harper (1902)