I am not a mathematician and I need your help about naming a relation holding between subsets that are in a particular relation to one another. (For the fact that I am not a mathematician and hence my question will very likely seem trivial to you -- I apologize, but I really do not have anybody to ask for help about this...).
I will use the symbol ">" to indicate containment. I need to name (and ideally also describe) the structure where $\alpha$ and $\beta$ are in a superset-subset relation just like their ingredients ($a,b$ and $x,y$, respectively) but at the same time only their bottom ingedients (that is, $b$ and $y$, respectively) are necessary ingredients for $\alpha$ and $\beta$ to exist (that's why I named $b$ and $y$ "proper subsets" in my original formulation of my question -- if I've done it in the wrong or a confusing way, I apologize, I have now modified it and hopefully made it clearer. I have also added an actual structure that I need to describe).
Assumptions:
$\alpha$ is $[a>b>\beta]$ and
$\beta$ is $[x>y]$
In the diagrams below, the arrow from $b$ (the bottom element of $\alpha$) indicates a pointer to $\beta$ (either its superset or subset).
I need to describe the relation that captures the following options to be true
... and which at the same time disallows the following options to be impossible:
So, back to my question: is there terminology or a descritive formulation which I can apply to the relation holding between $a,b,x$ and $y$? Thanks!
First of all, it is customary to denote "inclusion" in the sense of subsets with $\subseteq$ and $\subset$ for a proper subset instead of $>$. $A \subset B \subset C$ is often visualized as below.
I feel like the answer you are looking for is simply "transitivity", formally as given as
$$ \forall a, \forall b, \forall c: ( a \ast b ) \wedge ( b \ast c ) \rightarrow ( a \ast c )$$
where $a,b,c$ are sets, and $\ast$ is a relation. This means that, whenever the relation is true for $a$ and $b$, and for $b$ and $c$, then it is true for $a$ and $c$. It is true that set inclusion (that is, the subset relation $\subseteq$) is transitive, hence
$$ \forall a, \forall b, \forall c: ( a \subseteq b ) \wedge ( b \subseteq c ) \rightarrow ( a \subseteq c )$$
This is why in my example above $A \subset C$ and in yours $y \subset x \subset b \subset a$ or as you wrote $[[a>b] > [x >y]]$.
I am not sure what you mean by "the composition of the structure $[\alpha > \beta]$. If you worry about $\alpha$ and $\beta$, again, having subsets, this is nothing to worry about. The relation is still simply transitive, without the need to further specify like "2-dimensional" or "double transitive".
Also, as has been pointed out in the comments, most of your statements do not make sense (formally). If you say that $[a > b]$ is a statement, then by writing $[[a > b] >y]$, you write that $y$ is contained in the statement, which does not make sense. Alternatively, you could write $a > b >y$ which is short hand for $a > b \wedge b > y$, meaning that $b$ is contained in $a$ and that $y$ is contained in $b$.