# Total order

## Order whose elements are all comparable / From Wikipedia, the free encyclopedia

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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 $\leq$ on some set $X$, which satisfies the following for all $a,b$ and $c$ in $X$:

- $a\leq a$ (reflexive).
- If $a\leq b$ and $b\leq c$ then $a\leq c$ (transitive).
- If $a\leq b$ and $b\leq a$ then $a=b$ (antisymmetric).
- $a\leq b$ or $b\leq a$ (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.[1]
Total orders are sometimes also called **simple**,[2] **connex**,[3] or **full orders**.[4]

A set equipped with a total order is a **totally ordered set**;[5] the terms **simply ordered set**,[2] **linearly ordered set**,[3][5] and **loset**[6][7] are also used. The term *chain* is sometimes defined as a synonym of *totally ordered set*,[5] 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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