Strict total order

Remarks

The relation $<$ on $\R$ is a strict total order: it is irreflexive, transitive, and for distinct reals $x$ and $y$, $x < y$ or $y < x$. The relation $\subsetneq$ on the power set of $\set{1, 2}$ is a strict partial order but not a strict total order. The sets $\set{1}$ and $\set{2}$ are distinct, and neither is a proper subset of the other. The condition gives trichotomy: for all $a, b \in A$, exactly one of $a < b$, $a = b$, $b < a$ holds. The strict total orders on $A$ correspond one to one with the total orders on $A$, by $a \le b$ if and only if $a < b$ or $a = b$. Other names are strict linear order and strict simple order.