Toset
In mathematics, a total order 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}
:
a
≤
a
{\displaystyle a\leq a}
(reflexive).
If
a
≤
b
{\displaystyle a\leq b}
and
b
≤
c
{\displaystyle b\leq c}
then
a
≤
c
{\displaystyle a\leq c}
(transitive).
If
a
≤
b
{\displaystyle a\leq b}
and
b
≤
a
{\displaystyle b\leq a}
then
a
=
b
{\displaystyle a=b}
(antisymmetric).
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.
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