I worked this theory : A new generalization of the dimension?

For have a theorem of the dimension, and be more generaler and simpler than the Matroïds.

Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary intersections, with $X \in \mathcal T$.

Definition 2: Let $U \subset X$ we denote $\langle U\rangle_S=\bigcap \limits_{F \in \mathcal T, U \subset F} F$

Definition 3: $S=(X,\mathcal T)$ a structure, we say the set $U\neq \emptyset$ is free if :

$\forall u \in U, u \notin \langle v \mid v \in U,v\neq u \rangle_S$

Definition 4: $S=(X,\mathcal T)$ a structure, we say this structur have a dimension if:

$\forall U \subset X$ free, with $v \notin \langle U\rangle$, $U \cup \{v\}$ is free.

Definition 5: $S=(X,\mathcal T)$ a structure, $F \in \mathcal T$, we note $\dim(F)=n$ if:

the largest free set of $F$ have a cardinal of $n$.

Theorem: $S=(X,\mathcal T)$ a structure with a dimension, $E,F \in \mathcal T$,

if $\dim(E)=\dim(F)<\infty$ and $E \subset F$ then $E=F$

Question: Is this generalization of the dimension already existing?