  For a group $G$ and a finite set $A$, a cellular automaton (CA) is a
transformation $\tau : A^G \to A^G$ defined via a finite memory set $S
\subseteq G$ and a local map $\mu : A^S \to A$. Although memory sets are not
unique, every CA admits a unique minimal memory set, which consists on all the
essential elements of $S$ that affect the behavior of the local map. In this
paper, we study the links between the minimal memory set and the generating
patterns $\mathcal{P}$ of $\mu$; these are the patterns in $A^S$ that are not
fixed when the cellular automaton is applied. In particular, we show that when
$\vert S \vert \geq 2$ and $\vert \mathcal{P} \vert$ is not a multiple of
$\vert A \vert$, then the minimal memory set must be $S$ itself. Moreover, when
$\vert \mathcal{P} \vert = \vert A \vert$, $\vert S \vert \geq 3$, and the
restriction of $\mu$ to these patterns is well-behaved, then the minimal memory
set must be $S$ or $S \setminus \{s\}$, for some $s \in S \setminus \{e\}$.
These are some of the first general theoretical results on the minimal memory
set of a cellular automaton.
