**Definition**: Strict Total Order, Strictly Ordered Set

A total order “$\prec$”, in which the property of being reflexive is replaced by the property of being irreflexive is called a **strict total order** (or **trichotomous order**).

In a trichotomous order, for all elements \(a,b\in V\) exactly one of the following cases holds:

- Either $a \prec b$ ($a$ is smaller than $b$),
- or $a \succ b$ ($a$ is greater than $b$),
- or $a=b$ ($a$ equals $b$).

A chain $V$ with a strict total order is called a **strictly ordered set**. A strict total order is sometimes also called

| | | | | Contributors: *bookofproofs* | References: [979]

(none)

[979] **Reinhardt F., Soeder H.**: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10

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