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## 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]

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