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When S is finite, its completion is also finite, and has the smallest number of elements among all finite complete lattices containing S. [ 12 ] The partially ordered set S is join-dense and meet-dense in the Dedekind–MacNeille completion; that is, every element of the completion is a join of some set of elements of S , and is also the meet ...
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 ...
The best-known example is the existence of all suprema, which is in fact equivalent to the existence of all infima. Indeed, for any subset X of a poset, one can consider its set of lower bounds B . The supremum of B is then equal to the infimum of X : since each element of X is an upper bound of B , sup B is smaller than all elements of X , i.e ...
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.
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 family of all antichains in a finite partially ordered set can be given join and meet operations, making them into a distributive lattice. For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are called Sperner families and their lattice is a free distributive lattice , with a Dedekind ...
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. [1] The Dedekind number M ( n ) {\displaystyle M(n)} is the number of monotone Boolean functions of n {\displaystyle n} variables.
The previous example can be generalized to Dedekind domains. Let R be a Dedekind domain, K its field of fractions, and let P be a non-zero prime ideal of R. Then, the localization of R at P, denoted R P, is a principal ideal domain whose field of fractions is K.