# Minimal generating sets of monoids acting on finite vector spaces.

Let $V$ be a finite dimensional vector space over $\mathbb{Z}_2$ with a linear map $f_i : V \to V$ for each $i$ in some finite index set $I$.

Then one can always find some subset $G \subseteq V$ of minimal cardinality such that the set of all elements:

$f_{i_1} \circ \dots \circ f_{i_n}(g)$ where $g \in G$, $i_j \in I$ and $n \geq 0$

spans the vector space $V$. Likewise in the dual situation $(V^*,(f_i^*)_{i \in I})$ there is some respective minimal generating set $G' \subseteq V^*$.

Here are my questions:

1. How does the cardinality of $G'$ compare to that of $G$?

2. What if each $f_i$ is assumed to be idempotent?

Any help much appreciated.

-

Consider the dual module V*. It can be identified with the mappings $M\to F$ with right FM-module structure given by (fm)(x)=f(mx). In particular, if m is not the identity map, then fm is a constant map. Thus V*/constants is an n-dimensional module annihilated by all non-zero elements of M and hence cannot be generated by fewer than n elements (that is, a basis). Thus V* cannot be generated by fewer than n elements.