totally ordered
adj1
Mathematics, Order theory, Sciences, Set theory That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset.
- 1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67, (A.2.5) THEOREM If A is a totally ordered ring and if I is a proper order ideal, then A/I is a totally ordered ring (with the operations and order given above).
- The set {1, 2, 3, 4, 6, 8, 12, 24}, ordered by 'divides' is not totally ordered, since (for example) 6 and 8 are not related. However, the set {1, 2, 4, 12, 24} is totally ordered by the relation 'divides'.1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91:
1 more example
- G is a transitive group of order automorphisms of a totally ordered set X, θ is a convex G-congruence on the totally ordered set X, ξ is the order type of the totally ordered set X, ◌̅η is the order type of some class Y=xη of the congruence η,1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation
- Synonyms
- linearly ordered
- Derived terms
- totally ordered set
- Related terms
- chain