Total Strict Order Relation

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A Total Strict Order Relation is a Binary Relation that is a Transitive, a Antisymmetric and a Semiconnex Relation.



References

2020a

  • (Wikipedia, 2020a) ⇒ https://en.wikipedia.org/wiki/Total_order Retrieved:2020-2-15.
    • In mathematics, a total order, simple order, linear order, connex order, or full order is a binary relation on some set [math] X [/math] , which is antisymmetric, transitive, and a connex relation. A set paired with a total order is called a chain,a totally ordered set,a simply ordered set,or a linearly ordered set. Formally, a binary relation [math] \leq [/math] is a total order on a set [math] X [/math] if the following statements hold for all [math] a, b [/math] and [math] c [/math] in [math] X [/math] :
      • Antisymmetry: If [math] a \leq b [/math] and [math] b \leq a [/math] then [math] a = b [/math] ;
      • Transitivity: If [math] a \leq b [/math] and [math] b \leq c [/math] then [math] a \leq c [/math] ;
      • Connexity: [math] a \leq b [/math] or [math] b \leq a [/math] .
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2020b

  • (Wikipedia, 2020b) ⇒ https://en.wikipedia.org/wiki/Total_order#Strict_total_order Retrieved:2020-2-15.
    • For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) transitive semiconnex relation <, called a strict total order or strict semiconnex order,which can be defined in two equivalent ways:
      • a < b if ab and ab
      • a < b if not ba (i.e., < is the inverse of the complement of ≤)
    • Properties:
      • The relation is transitive: a < b and b < c implies a < c.
      • The relation is trichotomous: exactly one of a < b, b < a and a = b is true.
      • The relation is a strict weak order, where the associated equivalence is equality.
    • We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤ can be defined in two equivalent ways:
      • ab if a < b or a = b
      • ab if not b < a
    • Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.

      We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whether we are talking about the non-strict or the strict total order.