I came up with this problem just out of curiosity, thus I'm not really sure how to properly phrase/tag it, so my apologies for the clumsy expression.
Let say I have a set of objects $A=(a,b,c,\dots,n)$ observed over a period of time $T$. Each object is associated with some time intervals representing its existence duration, say the isomorphic set $I=(i_a,i_b,\dots,i_n)$, with $i_a=(t_{a1},t_{a2},\dots)$, and $t_{aj}$ representing some time intervals in $T$. Given a Boolean evaluation function $f:(A\times T)\to \{0,1\}$ on the existence of the object at certain time instance $t$ in $T$, can I constructing an isomorphism between $A$ and $I$ based on $f$? I'm thinking of something along the lines of the union of the inverses of $f^{-1}$, but not really sure how to write a consistent inverse of Cartesian product.
Edit (This example is in the comment, I'm just moving it here for better clarification): For example, I have 3 atoms (a,b,c) over a period t = (0, 5). I have a function f that tells me if these atoms exist at some time t1 or not (for example, f(a, 1) means at the time instance 1, atom a do exist). Now, I want to find some mapping g(a) based on f such that, for example, g(a) = ((0.5, 1.2), (1.5, 3), etc.).
I think I know what you mean in the example, but it is by no means an 'isomorphism between $A$ and $I$'.
Given $f:A\times T\to\{0,1\}$ you seem to be looking for $g:A\to \mathcal P(T)$ which assigns $$g(a)\ =\ \{t\in T:f(a,t)=1\}\,.$$ If you insist, you can write it in a less readable formula with an inverse of $f$: $$g(a)=\pi_T\left(f^{-1}(1)\,\cap\, \pi_A^{-1}(a)\right)$$ where $\pi_T,\pi_A$ are the projections from $A\times T$.
Or, define $f_a:=t\mapsto f(a,t)$ for each $a\in A$, and then we can write $$g(a)=f_a^{-1}(1)\,.$$ Note that it depends on $f$ whether these subsets of $T$ are indeed a union of disjoint intervals or not.