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 totality).
Requirements 1. to 3. just make up the definition of a partial order.
Reflexivity (1.) already follows from strong 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, toset and loset are also used. The term chain is sometimes defined as a synonym of totally ordered set, but generally refers to a totally ordered subset 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.
Wild Trip 2022
- 2022-04-24T00:00:00.000000Z
A-Game 2021
- 2021-03-02T00:00:00.000000Z
Mad Dazed 2021
- 2020-10-30T00:00:00.000000Z
Flat Beat 2020
- 2020-04-27T00:00:00.000000Z
Klein 2020
- 2019-12-13T00:00:00.000000Z
Similar Artists