Linearly ordered set

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set ${\displaystyle X}$, which is antisymmetric, transitive, and total (this relation is denoted here by infix ${\displaystyle \leq }$). A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.

More symbolically, a set ${\displaystyle X}$ is totally ordered under ${\displaystyle \leq }$ if the following statements hold for all ${\displaystyle a,b}$ and ${\displaystyle c}$ in ${\displaystyle X}$:

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