(definition)

**Definition:**
An order defined for all pairs of items of a set. For instance, ≤ (less than or equal to) is a total order on integers, that is, for any two integers, one of them is less than or equal to the other.

**Formal Definition:** A total order is a *relation* that is *reflexive*, *transitive*, *antisymmetric*, and total.

**Also known as** linear order.

**See also**
*partial order*, *chain*.

*Note:
Subset (⊆) is partial not total, since {a} is not a subset of {b}, nor is {b} a subset of {a}. The sets {a} and {b} are incomparable. *

* How could an order be total, but not transitive? If it is cyclic, like in the game Paper, Scissors, Rock.*

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Entry modified 11 February 2019.

HTML page formatted Wed Mar 13 12:42:46 2019.

Cite this as:

Paul E. Black and Paul J. Tanenbaum, "total order", in
*Dictionary of Algorithms and Data Structures* [online], Paul E. Black, ed. 11 February 2019. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/totalorder.html