Operations on Fuzzy Power sets

Question

With reference to a question:Operations on power set, I wanted to generalize it for a power set of fuzzy sets. So, first let us try to define power set of fuzzy sets. Let $\mathscr{F}(X)$ be the set of all fuzzy sets $A: X \longrightarrow \left[ 0, 1 \right]$ defined over $X$.

Now, considering the operations of fuzzy union $u(a, b)$ and fuzzy intersection $i(a, b)$, where for certain element $x$ of $X$, the membership value of $x$ in the fuzzy sets $A$ and $B$ are $A(x) = a$ and $B(x) = b$ respectively, which must satisfy the following properties:-

Fuzzy Union:-

1. The boundary conditions that $u(a, 0) = a$
2. The monotonicity that if $b \leq d$ then $u(a, b) \leq u(b, d)$
3. Commutativity that $u(a, b) = u(b, a)$
4. Associativity that $u(a, u(b, d)) = u(u(a, b), d)$

Fuzzy Intersection:-

1. The boundary conditions that $u(a, 1) = a$
2. The monotonicity that if $b \leq d$ then $u(a, b) \leq u(b, d)$
3. Commutativity that $u(a, b) = u(b, a)$
4. Associativity that $u(a, u(b, d)) = u(u(a, b), d)$

So, considering these properties, the following things are clear:-

1. The fuzzy power set $\mathscr{F}(X)$ of a crisp set $X$ is closed under both union and intersection.
2. The operations of union and intersection are associative (by definition).
3. The identities for both these operations exists and are (respectively for intersection and union) $X$ and $\phi_F$, where $X$ is a fuzzy set over $X$ defined as $\forall x \in X, X(x) = 1$ and $\phi_F$ is a fuzzy set over $X$ defined as $\forall x \in X, X(x) = 0$. This is a direct consequence of the boundary conditions in the definitions.

So, until now we can conclude that the fuzzy power set $\mathscr{F}(X)$ over a crisp set $X$ forms a monoid, infact a commutative monoid. And the arguments given in the Proof for crisp power set not being able to be a group can also be given here to prove that it does not form a group.

Now, the question is, after coming to this almost trivial result, in what all directions can I move further? This was a thought that came to my mind when I was travelling and once I was able to convince myself that this thought is absolutely correct, I am not able to think further. Any insights will be helpful for what to do next to extend this thought being in the fuzzy theory.