Comparability

Rigorous definition

A binary relation on a set ${\displaystyle P}$ is by definition any subset ${\displaystyle R}$ of ${\displaystyle P\times P.}$ Given ${\displaystyle x,y\in P,}$ ${\displaystyle xRy}$ is written if and only if ${\displaystyle (x,y)\in R,}$ in which case ${\displaystyle x}$ is said to be related to ${\displaystyle y}$ by ${\displaystyle R.}$ An element ${\displaystyle x\in P}$ is said to be ${\displaystyle R}$-comparable, or comparable (with respect to ${\displaystyle R}$), to an element ${\displaystyle y\in P}$ if ${\displaystyle xRy}$ or ${\displaystyle yRx.}$ Often, a symbol indicating comparison, such as ${\displaystyle <}$ (or ${\displaystyle \le ,}$ ${\displaystyle >,}$ ${\displaystyle \ge ,}$ and many others) is used instead of ${\displaystyle R,}$ in which case ${\displaystyle x<y}$ is written in place of ${\displaystyle xRy,}$ which is why the term "comparable" is used.

Comparability with respect to ${\displaystyle R}$ induces a canonical binary relation on ${\displaystyle P}$; specifically, the comparability relation induced by ${\displaystyle R}$ is defined to be the set of all pairs ${\displaystyle (x,y)\in P\times P}$ such that ${\displaystyle x}$ is comparable to ${\displaystyle y}$; that is, such that at least one of ${\displaystyle xRy}$ and ${\displaystyle yRx}$ is true. Similarly, the incomparability relation on ${\displaystyle P}$ induced by ${\displaystyle R}$ is defined to be the set of all pairs ${\displaystyle (x,y)\in P\times P}$ such that ${\displaystyle x}$ is incomparable to ${\displaystyle y;}$ that is, such that neither ${\displaystyle xRy}$ nor ${\displaystyle yRx}$ is true.

If the symbol ${\displaystyle <}$ is used in place of ${\displaystyle \le }$ then comparability with respect to ${\displaystyle <}$ is sometimes denoted by the symbol ${\displaystyle \stackrel{<}{\underset{>}{=}}}$, and incomparability by the symbol ${\displaystyle \cancel{\stackrel{<}{\underset{>}{=}}}}$. Thus, for any two elements ${\displaystyle x}$ and ${\displaystyle y}$ of a partially ordered set, exactly one of ${\displaystyle x\stackrel{<}{\underset{>}{=}}y}$ and ${\displaystyle x\cancel{\stackrel{<}{\underset{>}{=}}}y}$ is true.