* Sperner's lemma

Sperner's lemma, from 1928, states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors.

** Sperner's lemma

In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it.

* Sperner's lemma

It is sometimes called Sperner's lemma, but that name also refers to another result on coloring.

By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point ( See also Sperner's lemma ).

Sperner's theorem ( a special case of Dilworth's theorem ) states that these families are the largest possible Sperner families over an n-set.

A graded partially ordered set is said to have the Sperner property when one of its largest antichains is formed by a set of elements that all have the same rank ; Sperner's theorem states that the poset of all subsets of a finite set, partially ordered by set inclusion, has the Sperner property.

To differentiate the two results, the result on the size of a Sperner family is now more commonly known as Sperner's theorem.

: You may be looking for Sperner's theorem on set families

The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory.

Sperner's students included Kurt Leichtweiss and Gerhard Ringel.

In one dimension, Sperner's Lemma can be regarded as a discrete version of the Intermediate Value Theorem.

Sperner's Lemma can be used to get as close an approximation as desired to an envy-free solutions for many players.

This inequality has many applications in combinatorics ; in particular, it can be used to prove Sperner's theorem.

using Sperner's theorem.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i. e. given any a and b, with a > b, there exist c and d, all positive and rational, such that

In the theory of formal languages in computability theory, a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of times, with the resulting string remaining in that language.

In this case, the covariant version of Yoneda's lemma states that

The lemma states that, under certain conditions, an event will occur with probability zero or with probability one.

The Borel – Cantelli lemma states:

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.

Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.

Zorn's lemma, also known as the Kuratowski – Zorn lemma, is a proposition of set theory that states:

We can prove the cancellation law easily using Euclid's lemma, which generally states that if an integer b divides a product rs ( where r and s are integers ), and b is relatively prime to r, then b must divide s. Indeed, the equation

The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism.

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent.

The form of König's lemma most convenient for this purpose is the one which states that any infinite finitely branching subtree of < math >

In the mathematical theory of queues, Little's result, theorem, lemma, law or formula is a theorem by John Little which states:

In its simplest form, Itō's lemma states the following: for an Itō drift-diffusion process

In higher dimensions, Ito's lemma states

Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element.

The ultrafilter lemma states that every filter on a set is contained within some maximal ( proper ) filter — an ultrafilter.

Conversely, a complete variety is close to being a projective variety: Chow's lemma states that if X is a complete variety, there is a projective variety Z and a birational morphism Z → X.

The diagonal lemma states that there is a sentence φ such that φ ↔ ψ (< u >#( φ )</ u >) is provable in T.

In mathematics, Dickson's lemma states that every set of-tuples of natural numbers has finitely many minimal elements.

* The Morse lemma states that non-degenerate critical points of certain functions are isolated.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring ; every division ring arises in this fashion from some simple module.

Euclid's classical lemma can be rephrased as " in the ring of integers every irreducible is prime ".

The canonical model is a model of L, as every L-MCS contains all theorems of L. By Zorn's lemma, each L-consistent set is contained in an L-MCS, in particular every formula unprovable in L has a counterexample in the canonical model.

We will go over a typical application of Zorn's lemma: the proof that every nontrivial ring R with unity contains a maximal ideal.

For instance, the third proof uses that every filter is contained in an ultrafilter ( i. e., a maximal filter ), and this is seen by invoking Zorn's lemma.

Zorn's lemma is also used to prove Kelley's theorem, that every net has a universal subnet.

The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology.

The question of points was close to resolution by 1950 ; Alexander Grothendieck took a sweeping step ( appealing to the Yoneda lemma ) that disposed of it — naturally at a cost, that every variety or more general scheme should become a functor.

This lemma is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn – Banach theorem and Tychonoff's theorem, the existence of a Hamel basis for every vector space, and the existence of an algebraic closure for every field.

The diagonal lemma also requires that there be a systematic way of assigning to every formula θ a natural number #( θ ) called its Gödel number.

Under this correspondence, Dickson's lemma may be seen as a special case of Hilbert's basis theorem stating that every polynomial ideal has a finite basis, for the ideals generated by monomials.

On a contractible domain, every closed form is exact by the Poincaré lemma.

It follows from the handshaking lemma, proven by Leonhard Euler in 1736 as part of the first paper on graph theory, that every cubic graph has an even number of vertices.

Zorn's lemma states that, if a poset has an upper bound for every chain, then it has a maximal element.