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Dushnik–Miller theorem

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Not to be confused withErdős–Dushnik–Miller theorem.

In mathematics, theDushnik–Miller theorem is a result inorder theory stating that everycountably infinite linear order has a non-identityorder embedding into itself.^[1] It is named for Ben Dushnik and E. W. Miller, who proved this result in a paper of 1940; in the same paper, they showed that the statement does not always hold foruncountable linear orders, using theaxiom of choice to build a suborder of thereal line ofcardinality continuum with no non-identity order embeddings into itself.^[2]

Inreverse mathematics, the Dushnik–Miller theorem for countable linear orders has the same strength as thearithmetical comprehension axiom (ACA₀), one of the "big five" subsystems ofsecond-order arithmetic.^[1]^[3] This result is closely related to the fact that (asLouise Hay and Joseph Rosenstein proved) there existcomputable linear orders with no computable non-identity self-embedding.^[3]^[4]

References

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^^a ^bDowney, Rodney G.;Jockusch, Carl; Miller, Joseph S. (2006), "On self-embeddings of computable linear orderings",Annals of Pure and Applied Logic,138 (1–3):52–76,doi:10.1016/j.apal.2005.06.008,MR 2183808
^Dushnik, Ben; Miller, E. W. (1940), "Concerning similarity transformations of linearly ordered sets",Bulletin of the American Mathematical Society,46 (4):322–326,doi:10.1090/S0002-9904-1940-07213-1,MR 0001919
^^a ^bHirschfeldt, Denis R. (2014), "10.1 The Dushnik–Miller theorem",Slicing the Truth, Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore, vol. 28, World Scientific
^Rosenstein, Joseph G. (1982),Linear Orderings, Pure and Applied Mathematics, vol. 98, Academic Press,Theorem 16.49, p. 447,ISBN 0-12-597680-1,MR 0662564

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology &Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

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