# 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.

**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] .

- In mathematics, a

- (...)

### 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.

- The relation is transitive:
- 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.

- For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) transitive semiconnex relation <, called a