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This definition of "infinite set" should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form {0, 1, 2, ..., n−1} for some natural number n – an infinite set is one that is literally "not finite", in the sense of bijection.
Several researchers have investigated algorithms for constructing the Dedekind–MacNeille completion of a finite partially ordered set. The Dedekind–MacNeille completion may be exponentially larger than the partial order it comes from, [12] and the time bounds for such algorithms are generally stated in an output-sensitive way, depending ...
The ring = of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal I of R, R/I is a finite set, and recall that a finite integral domain is a field; so by (DD4) R is a Dedekind domain. As above, this includes all the examples considered ...
The existence of such a set is consistent, for example given in Cohen's first model. [8] Surprisingly, however, being an infinite Dedekind-finite set is not enough to rule out a group structure, as it is consistent that there are infinite Dedekind-finite sets with Dedekind-finite power sets. [9]
If the ordered set S is complete, then, for every Dedekind cut (A, B) of S, the set B must have a minimal element b, hence we must have that A is the interval (−∞, b), and B the interval [b, +∞). In this case, we say that b is represented by the cut (A, B). The important purpose of the Dedekind cut is to work with number sets that are not ...
In general, such a J may not exist and consequently the set of ideal classes of R may only be a monoid. However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R.
When interpreted as a proof within a first-order set theory, such as ZFC, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory ...
The fiber of a Néron model over a closed point of Spec(R) is a smooth commutative algebraic group, but need not be an abelian variety: for example, it may be disconnected or a torus. Néron models exist as well for certain commutative groups other than abelian varieties such as tori, but these are only locally of finite type.