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A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. [1] A simple example is , the set of natural numbers.
The Dedekind–MacNeille completion of S has the same order dimension as does S itself. [19] In the category of partially ordered sets and monotonic functions between partially ordered sets, the complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of S is the injective hull of S. [20]
One application is to give factorisations of Dedekind zeta-functions, for example in the case of a number field that is Galois over the rational numbers. In accordance with the decomposition of the regular representation into irreducible representations , such a zeta-function splits into a product of Artin L -functions, for each irreducible ...
The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. [2] It can be used to prove many of the fundamental results of real analysis , such as the intermediate value theorem , the Bolzano–Weierstrass theorem , the extreme value theorem , and the Heine ...
is a finite set. (Richard Dedekind) Every one-to-one function from into itself is onto. A set with this property is called Dedekind-finite. Every surjective function from onto itself is one-to-one. is empty or every partial ordering of contains a maximal element.
For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field () with x 3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does.
A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; that is, every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective. [3]
The Dedekind-finite property is characterized. Most significantly, representation of P 1 (R) in a projective space over a division ring K is accomplished with a (K, R)-bimodule U that is a left K-vector space and a right R-module. The points of P 1 (R) are subspaces of P 1 (K, U × U) isomorphic to their complements.