Example 1

Every field extension is an inflation. Let indeed K be an extension of a field k. Let us furnish K with a well-ordering for which 0 is the greatest element, and for $\lambda \in K$, let $K_\lambda$ be the sub-$k$-extension of K generated by the elements $\beta$ of K such that $\beta < \lambda$. The verification of conditions a), b), c), for $k, K$ and the family $(K_\lambda)_{\lambda \in K}$, is immediate.