Total Strict Order Relation

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

• AKA: Total Strict Partial Order Relation, Strict Total Order Relation, Strict Semiconnex Order, Strict Semiconnex Order, Linear Order, Total Order, Full Order, Simple Order, Connex Order.
• Context:
  • It is associated to Totally Ordered Set, Ordered field, Ordered Vector Space and Totally Ordered Group.
• Example(s):
  • a GreaterThan Relation associated to The Number Line;
  • alphabet letters ordered by the standard dictionary order;
  • any subset of a totally ordered set $X$ for the restriction of the order on $X$;
  • any set of cardinal numbers or ordinal numbers,
  • Lexicographical Order on the Cartesian Product of a family of totally ordered sets, indexed by a well ordered set;
  • a Finite Total Order.
  • …
• Counter-Example(s):
  • a GreaterThanOrEqualTo Relation associated to The Number Line,
  • a Partial Order,
  • a Total Weak Order Relation,
  • a Total Preorder Relation,
  • a Total Partial Order Relation.
• See: Ordered Set, Partially Ordered Set, Order Theory, Infix Notation, Set (Mathematics), Logical Disjunction, Comparability, Reflexive Relation, Linear Extension, Non-Total Strict Order Relation, Non-Total Order Relation, Lexicographical Order, Product Order, Direct Product, Cartesian Product, Order Completeness, Order Topology, Category Theory, Lattice Theory.

---

---

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]\displaystyle{ 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]\displaystyle{ \leq }[/math] is a total order on a set [math]\displaystyle{ X }[/math] if the following statements hold for all [math]\displaystyle{ a, b }[/math] and [math]\displaystyle{ c }[/math] in [math]\displaystyle{ X }[/math] :
    • Antisymmetry: If [math]\displaystyle{ a \leq b }[/math] and [math]\displaystyle{ b \leq a }[/math] then [math]\displaystyle{ a = b }[/math] ;
    • Transitivity: If [math]\displaystyle{ a \leq b }[/math] and [math]\displaystyle{ b \leq c }[/math] then [math]\displaystyle{ a \leq c }[/math] ;
    • Connexity: [math]\displaystyle{ a \leq b }[/math] or [math]\displaystyle{ b \leq a }[/math] .

(...)

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 a ≤ b and a ≠ b
    • a < b if not b ≤ a (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:
    • a ≤ b if a < b or a = b
    • a ≤ b 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.

---