Knaster-Tarski Theorem says that for a function $f\colon L\to L$, where $L$ is a complete lattice, and $f$ is a increasing function, then a $f$ has a fixed point.
Question:
For a function $f\colon \mathbb{R} \to \mathbb{R}$, which is defined by $f(x)= x+b$, where $b\neq 0$. $f$ does not have a fixed point, which appears a contradiction to the Knaster-Tarski Theorem.
Where am I missing?
Thank you in advance
The fact that $\Bbb R$ is not a complete lattice since a complete lattice must have minimum and maximum. It is worth noting that in this context increasing means $x\leq f(x)$; rather than $x\leq y\rightarrow f(x)\leq f(y)$.
On the other hand, any function $f\colon[0,1]\to[0,1]$ such that $x\leq f(x)$, has a fixed point (e.g. $1$ has to be a fixed point under this condition).