A space whose topology is an order topology is called a Linearly Ordered Topological Space ( LOTS ), and a subspace of a linearly ordered topological space is called a Generalized Ordered Space ( GO-space ).
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Another historical turnpoint can be traced back to 1895, when Georg Cantor proved in a theorem that if a binary relation is " linearly ordered ", then it is also isomorphically embeddable in the ordered real numbers.
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Dependence logic is a game theoretic semantics can be obtained from that of first-order logic by restricting the availability of information to the players, thus allowing for non-linearly ordered patterns of dependence and independence between variables.
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The result follows fairly easily because it is easy to see that the non-standard numbers have to be dense and linearly ordered without endpoints, and the order type of the rationals is the only countable dense linear order without endpoints.
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For example, the subdirect irreducibles in the variety generated by a finite linearly ordered Heyting algebra " H " must be just the nondegenerate quotients of " H ", namely all smaller linearly ordered nondegenerate Heyting algebras.
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For example, the subdirect irreducibles in the variety generated by a finite linearly ordered Heyting algebra " H " must be just the nondegenerate quotients of " H ", namely all smaller linearly ordered nondegenerate Heyting algebras.
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The observation that large cardinal axioms are linearly ordered by consistency strength is just that, an observation, not a theorem . ( Without an accepted definition of large cardinal property, it is not subject to proof in the ordinary sense ).
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:" Law of Finite Chains Between Events in the Combined Ordering " : There are no infinite chains ( " i . e . ", linearly ordered sets ) of events between two events in the combined ordering ?!.
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K�the's paper also required a uniserial ring to have a unique composition series, which not only forces the right and left ideals to be linearly ordered, but also requires that there be only finitely many ideals in the chains of left and right ideals.
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Examples of transitive relations include the equality relation on any set, the " less than or equal " relation on any linearly ordered set, and the relation " " x " was born before " y " " on the set of all people.
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