Consider 3 sets
$$
A\equiv \{a_1,a_2,a_3\}
$$
$$
B\equiv \{b_1,b_2,b_3\}
$$
$$
C\equiv \{c_1,c_2,c_3\}
$$
Let $\mathcal{A}$ denote all possible subsets of $A$ excluding the empty set. Similarly, $\mathcal{B}$ and $\mathcal{C}$. I.e.,
$$
\mathcal{A}\equiv \Big\{A, \{a_1\}, \{a_2\}, \{a_3\}, \{a_1,a_2\}, \{a_2, a_3\}, \{a_1,a_3\}\Big\}
$$
$$
\mathcal{B}\equiv \Big\{B, \{b_1\},\{b_2\}, \{b_3\},\{b_1,b_2\}, \{b_2, b_3\}, \{b_1,b_3\}\Big\}
$$
$$
\mathcal{C}\equiv \Big\{C, \{c_1\},\{c_2\}, \{c_3\},\{c_1,c_2\}, \{c_2, c_3\}, \{c_1,c_3\}\Big\}
$$
Some additional definitions are now introduced.
Construction of $A\times B \times C$:
Let $\times$ denote the cartesian product.
$A\times B \times C$ is the collection of all ordered $3$-tuples from $A,B,C$, i.e.,
$$
A\times B \times C\equiv \Big\{(a_1,b_1,c_1),(a_1,b_2,c_1), (a_1,b_3,c_1),… \Big\}
$$
Construction of $\mathcal{T}$:
Consider $\mathcal{T}\equiv \mathcal{A}\times \mathcal{B}\times \mathcal{C}$. For example, an element $T\in \mathcal{T}$ is
$$
T\equiv \Big(\{a_1,a_2\}, \{b_3\} , \{c_1,c_2\}\Big)
$$
Each $T\in \mathcal{T}$ is made of 3 "components". In the example above, the first component is $\{a_1,a_2\}$, the second is $\{b_3\}$, the third is $\{c_1,c_2\}$.
We rewrite each $T\in \mathcal{T}$ as the set of all possible ordered $3$-tuples that can be constructed from its $3$ components. Continuing the example above,
$$
T\equiv \overbrace{\Big(\{a_1,a_2\}, \{b_3\} , \{c_1,c_2\}\Big)}^{\text{Representation 1}} \equiv \overbrace{\Big\{\overbrace{(a_1,b_3, c_1)}^{\text{ordered $3$-tuple}}, (a_2, b_3, c_1), (a_1,b_3, c_2), (a_2,b_3, c_2)\Big\} }^{\text{Representation 2}}
$$
Construction of $\mathcal{M}$:
We collect in $\mathcal{M}$ all the subsets of $A\times B\times C$ that cannot be part of $\mathcal{T}$. For example,
$$
M\equiv \Big\{(a_1,b_3, c_1), (a_2, b_2, c_1)\Big\}
$$
$M$ is not an element of $\mathcal{T}$.
Question
It seems to be that the elements of $\mathcal{M}$ can be "categorised" into 2 separate groups: let $M$ denote a generic element of $\mathcal{M}$; let $M^c$ denote the complement of $M$ in $A\times B\times C$; we can have
(1) $M$ such that $M^c \in \mathcal{T}$. For example, $M\equiv A\times B\times C \setminus \Big\{(a_1,b_2,c_3)\Big\}$ where $A\times B\times C \setminus \Big\{(a_1,b_2,c_3)\Big\}$ is the complement of $\Big\{(a_1,b_2,c_3)\Big\}$ in $A\times B\times C$.
(2) $M$ such that $M^c\notin \mathcal{T}$. For example, $M\equiv A\times B\times C \setminus \Big\{(a_1,b_3,c_1), (a_2, b_2, c_3)\Big\}$.
My question is about group (2) and how I can relate a set in group (2) to $\mathcal{T}$.
More precisely: is there a way to rewrite an $M$ following in group (2) as if it was a cartesian product of sets? Or, express it in a way that relates it to the elements of $\mathcal{T}$? For group (1), I am fine since I have $M^c\in \mathcal{T}$. For group (2), I am looking for a relation with the elements in $\mathcal{T}$.
Best Answer
This answer is far from being complete (due to obvious reasons).
For finite $A, B, C$ there is a multitude of choices to represent a subset of $A \times B \times C$ through $A' \times B' \times C'$ with $A' \subseteq A$, $B' \subseteq B$ and $C' \subseteq C$ (the most trivial with $A', B', C'$ being singletons, of course). So you may need some criteria to choose "this over that". It may include the set operations allowed, their respective costs, the costs of selecting a particular subset of $A$ or $B$ or $C$, and whatever else.
Even just counting set-theoretic operations (union, intersection and complement) to reach a given subset of $A \times B \times C$ out of products of subsets of $A, B, C$ presents a fairly hard computational problem. With not too large sets, it can of course be solved using dynamic programming, and there are other methods, but don't expect too much. For instance, with $A = B = C = \{1, \ldots, n\}$ and $M = \{(x, y, z) : x + y + z \leqslant n\}$, you will still need $O(n^2)$ elements of $\mathcal{T}$ to get $M$.
So the question to you is what is the "measure of quality" (or complexity) of relations.
With infinite sets, counting operations is out of question. At the least, you enter the area of functional analysis, and at not the worst, you lose an ability to represent your $M$ exactly (from a computational point of view).
(If a representation of $M$ using singletons is all you need, then forget all of the above. Joking.)