{"history":[],"article":{"id":"db500ec2-f360-4a9c-bf8b-f64719c34060","title":"Dense order","slug":"dense-order","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a [partial order](https://bliptext.com/articles/partial-order) or [total order](https://bliptext.com/articles/total-order) < on a [set](https://bliptext.com/articles/set-mathematics) X is said to be **dense** if, for all x and y in X for which x < y, there is a z in X such that x < z < y. That is, for any two elements, one less than the other, there is another element between them.\nFor total orders this can be simplified to \"for any two distinct elements, there is another element between them\", since all elements of a total order are [comparable](https://bliptext.com/articles/comparability).\n\n# Example\nThe [rational numbers](https://bliptext.com/articles/rational-number) as a linearly ordered set are a densely ordered set in this sense, as are the [algebraic numbers](https://bliptext.com/articles/algebraic-number), the [real numbers](https://bliptext.com/articles/real-number), the [dyadic rationals](https://bliptext.com/articles/dyadic-rational) and the [decimal fractions](https://bliptext.com/articles/decimal-fraction).\nIn fact, every [Archimedean](https://bliptext.com/articles/archimedean-property) ordered [ring extension](https://bliptext.com/articles/subring) of the [integers](https://bliptext.com/articles/integers) is a densely ordered set.\nOn the other hand, the linear ordering on the [integers](https://bliptext.com/articles/integer) is not dense.\n\n# Uniqueness for total dense orders without endpoints\n[Georg Cantor](https://bliptext.com/articles/georg-cantor) proved that every two non-empty dense totally ordered [countable sets](https://bliptext.com/articles/countable-set) without lower or upper bounds are [order-isomorphic](https://bliptext.com/articles/order-isomorphism).\nThis makes the theory of dense linear orders without bounds an example of an ω-[categorical theory](https://bliptext.com/articles/categorical-theory) where ω is the smallest [limit ordinal](https://bliptext.com/articles/limit-ordinal).\nFor example, there exists an order-isomorphism between the [rational numbers](https://bliptext.com/articles/rational-number) and other densely ordered countable sets including the [dyadic rationals](https://bliptext.com/articles/dyadic-rational) and the [algebraic numbers](https://bliptext.com/articles/algebraic-number).\nThe proofs of these results use the [back-and-forth method](https://bliptext.com/articles/back-and-forth-method).\n[Minkowski's question mark function](https://bliptext.com/articles/minkowski-s-question-mark-function) can be used to determine the order isomorphisms between the [quadratic algebraic numbers](https://bliptext.com/articles/quadratic-irrational-number) and the [rational numbers](https://bliptext.com/articles/rational-number), and between the rationals and the [dyadic rationals](https://bliptext.com/articles/dyadic-rational).\n\n# Generalizations\nAny [binary relation](https://bliptext.com/articles/binary-relation) R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y are R-related.\nFormally:\n[Sufficient conditions](https://bliptext.com/articles/sufficient-condition) for a binary relation R on a set X to be dense are:\nNone of them are [necessary](https://bliptext.com/articles/necessary-condition).\nFor instance, there is a relation R that is not reflexive but dense.\nA [non-empty](https://bliptext.com/articles/empty-relation) and dense relation cannot be [antitransitive](https://bliptext.com/articles/antitransitive).\nA strict partial order < is a dense order [if and only if](https://bliptext.com/articles/if-and-only-if) < is a dense relation.\nA dense relation that is also [transitive](https://bliptext.com/articles/transitive-relation) is said to be [idempotent](https://bliptext.com/articles/idempotent-relation).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T08:13:22.690Z","updated_at":"2025-01-26T08:13:22.690Z","search_vector":"'dens':1 'order':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a [partial order](https://bliptext.com/articles/partial-order) or [total order](https://bliptext.com/articles/total-order) < on a [set](https://bliptext.com/articles/set-mathematics) X is said to be **dense** if, for all x and y in X for which x < y, there is a z in X such that x < z < y. That is, for any two elements, one less than the other, there is another element between them.\nFor total orders this can be simplified to \"for any two distinct elements, there is another element between them\", since all elements of a total order are [comparable](https://bliptext.com/articles/comparability).\n\n# Example\nThe [rational numbers](https://bliptext.com/articles/rational-number) as a linearly ordered set are a densely ordered set in this sense, as are the [algebraic numbers](https://bliptext.com/articles/algebraic-number), the [real numbers](https://bliptext.com/articles/real-number), the [dyadic rationals](https://bliptext.com/articles/dyadic-rational) and the [decimal fractions](https://bliptext.com/articles/decimal-fraction).\nIn fact, every [Archimedean](https://bliptext.com/articles/archimedean-property) ordered [ring extension](https://bliptext.com/articles/subring) of the [integers](https://bliptext.com/articles/integers) is a densely ordered set.\nOn the other hand, the linear ordering on the [integers](https://bliptext.com/articles/integer) is not dense.\n\n# Uniqueness for total dense orders without endpoints\n[Georg Cantor](https://bliptext.com/articles/georg-cantor) proved that every two non-empty dense totally ordered [countable sets](https://bliptext.com/articles/countable-set) without lower or upper bounds are [order-isomorphic](https://bliptext.com/articles/order-isomorphism).\nThis makes the theory of dense linear orders without bounds an example of an ω-[categorical theory](https://bliptext.com/articles/categorical-theory) where ω is the smallest [limit ordinal](https://bliptext.com/articles/limit-ordinal).\nFor example, there exists an order-isomorphism between the [rational numbers](https://bliptext.com/articles/rational-number) and other densely ordered countable sets including the [dyadic rationals](https://bliptext.com/articles/dyadic-rational) and the [algebraic numbers](https://bliptext.com/articles/algebraic-number).\nThe proofs of these results use the [back-and-forth method](https://bliptext.com/articles/back-and-forth-method).\n[Minkowski's question mark function](https://bliptext.com/articles/minkowski-s-question-mark-function) can be used to determine the order isomorphisms between the [quadratic algebraic numbers](https://bliptext.com/articles/quadratic-irrational-number) and the [rational numbers](https://bliptext.com/articles/rational-number), and between the rationals and the [dyadic rationals](https://bliptext.com/articles/dyadic-rational).\n\n# Generalizations\nAny [binary relation](https://bliptext.com/articles/binary-relation) R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y are R-related.\nFormally:\n[Sufficient conditions](https://bliptext.com/articles/sufficient-condition) for a binary relation R on a set X to be dense are:\nNone of them are [necessary](https://bliptext.com/articles/necessary-condition).\nFor instance, there is a relation R that is not reflexive but dense.\nA [non-empty](https://bliptext.com/articles/empty-relation) and dense relation cannot be [antitransitive](https://bliptext.com/articles/antitransitive).\nA strict partial order < is a dense order [if and only if](https://bliptext.com/articles/if-and-only-if) < is a dense relation.\nA dense relation that is also [transitive](https://bliptext.com/articles/transitive-relation) is said to be [idempotent](https://bliptext.com/articles/idempotent-relation).","revisionCount":0},"hasMore":false,"nextCursor":null}

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