Properties of the rank function on independence systems

Let's take the simplest counterexample to the exchange property. We want independent sets $A$ and $B$ such that $|A|>|B|$, but there is no $x\in A\setminus B$ such that $B\cup \{x\}$ is independent. So let $X = \{1,2,3\}$, $A = \{1,2\}$ and $B = \{3\}$, and $\mathcal{I} = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}\}$.

This gives an example where $\sigma_\rho$ is not increasing. Let $C = \{3\}$ and $C' = \{1,3\}$. We have $\rho(C) = \rho(C\cup \{1\}) = \rho(C\cup \{2\}) = 1$, so $\sigma_\rho(C) = \{1,2,3\}$. But $\rho(C') = 1$, while $\rho(C'\cup \{2\}) = 2$, so $\sigma_\rho(C') = C' = \{1,3\}$. Thus $\sigma_\rho$ is not increasing.