Preorder and Posets

Poset

A poset is a set $A$ equipped with a binary relation $a{\le }_{A}b$ such that the following conditions hold for all $a,b,c\in A$:

• reflexivity: $a{\le }_{A}a$
• transitivity: if $a{\le }_{A}b$ and $b{\le }_{A}c$, then $a{\le }_{A}c$
• antisymmetry: if $a{\le }_{A}b$ and $b{\le }_{A}a$, then $a=b$.

e.g. The real numbers $\mathbb{R}$ with their usual ordering $x\le y$ form a partially ordered set that is also linearly ordered: either $x\le y$ or $y\le x$ for any $x,y$.

Definition

Def Monotone Poset Function A function $m:A\to B$ with posets $A$ and $B$ is monotone if $a{\le }_{A}b⟹m(a){\le }_{B}m(b)$