(set theory) A proposition of set theory stating that every partially ordered set, in which every chain (i.e. totally ordered subset) has an upper bound, contains at least one maximal element.

(category theory) Given a category 𝒞 with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from 𝒞 to Sets, then there is a natural isomorphism between Nat(H,F), the set of natural transformations from H to F, and the set F(A). (Any natural transformation 𝛼 from H to F is determined by what 𝛼_A( mbox id_A) is.)

(number theory) The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;

Synonym of Teichmüller-Tukey lemma

Synonym of Bézout's identity

(mathematics) A lemma, of importance in harmonic analysis and asymptotic analysis, stating that the Fourier transform or Laplace transform of an L1 function vanishes at infinity.

(statistics) A lemma stating that when performing a hypothesis test between two point hypotheses H₀: θ = θ₀ and H₁: θ = θ₁, then the likelihood-ratio test which rejects H₀ in favour of H₁ when 𝛬(x)=(L(𝜃₀∣x))/(L(𝜃₁∣x))≤𝜂 where P(𝛬(X)≤𝜂∣H_0)=𝛼 is the most powerful test of size α for a threshold η.

(mathematics) A lemma stating that every non-empty collection of finite character has a maximal element with respect to inclusion.