total ordering - Definition. Was ist total ordering
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Was (wer) ist total ordering - definition

ORDERING RELATION WHERE ALL ELEMENTS CAN BE COMPARED, EQUALITY MEANS IDENTITY; BINARY RELATION ON SOME SET, WHICH IS ANTISYMMETRIC, TRANSITIVE, AND TOTAL
Infinite descending chain; TotalOrderedSet; Total ordered set; Totally ordered set; Linear order; Total ordering relation; Total ordering; Linearly ordered set; Totally ordered; Linear ordering; Linearly ordered; Chain (order theory); Total (order theory); Toset; Linear (order); Infinite descending chains; Strict total order; Totally-ordered set; Finite chain; Strict linear order; Finite total order; Simple order; Simply ordered set; Complete total order; Chain (ordered set); Complete ordering; Complete order; Ascending chain; Loset; Chain (poset)

total ordering         
<mathematics> A relation R on a set A which is a {partial ordering}; i.e. it is reflexive (xRx), transitive (xRyRz => xRz) and antisymmetric (xRyRx => x=y) and for any two elements x and y in A, either x R y or y R x. See also equivalence relation, well-ordered. (1995-02-16)
Total order         
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
totally ordered set         
<mathematics> A set with a total ordering.

Wikipedia

Total order

In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :

  1. a a {\displaystyle a\leq a} (reflexive).
  2. If a b {\displaystyle a\leq b} and b c {\displaystyle b\leq c} then a c {\displaystyle a\leq c} (transitive).
  3. If a b {\displaystyle a\leq b} and b a {\displaystyle b\leq a} then a = b {\displaystyle a=b} (antisymmetric).
  4. a b {\displaystyle a\leq b} or b a {\displaystyle b\leq a} (strongly connected, formerly called total).

Reflexivity (1.) already follows from connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders.

A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term chain is sometimes defined as a synonym of totally ordered set, but refers generally to some sort of totally ordered subsets of a given partially ordered set.

An extension of a given partial order to a total order is called a linear extension of that partial order.