Define an operator $\mathop{\vec{\bigcup}}$ as follows:

Definition.Whenever $A$ is an $I$-indexed family of sets, where $I$ is a totally-ordered set, we have $$\mathop{\vec{\bigcup}}_{i \in I} A_i = \bigcup_{j<i} A_j$$

For example,

$$\mathop{\vec{\bigcup}}_{i \in \mathbb{N}} \{i\} = \{0,\ldots,i-1\}.$$

Convention.For the purposes of this question, lets refer to $\mathop{\vec{\bigcup}}$ as theserializationof the operator $\bigcup$.

We can serialize other operators, too, of course, using the exact same definition as above. For example:

$$\mathop{\vec{\sum}}_{i \in \mathbb{N}} i = \frac{1}{2} i(i-1)$$

And of course, this features in the definition of an infinite sum:

$$\sum_{i \in \mathbb{N}} a_i = \lim_{i \rightarrow \infty} \mathop{\vec{\sum_{i \in \mathbb{N}}}}a_i$$

In this context, the entity $\mathop{\vec{\sum_{i \in \mathbb{N}}}}$ could be referred to as the 'partial sum operator.'

Something quite similar seems to happen in calculus; when we write something like $$\int_5 2x dx = x^2-5^2$$ we're using the ordered structure of the real line, together with our ability to integrate over subsets of the real line, together with a distinguished basepoint, namely $5$, to "serialize" the Riemann integral (or Lebesgue integral, for that matter) with respect to $5$. FTC then tells us that, if certain hypothesize are met, the serialization of the Riemann integral is an inverse to differentiation.

Indeed, backing up a bit, lets define that:

Definition.If $A$ is an $\mathbb{N}$-indexed family of sets, then$$\mathop{\mathrm{disj}}_{i \in \mathbb{N}}A_i = A_i \setminus \mathop{\vec{\bigcup}}_{i \in I} A_i.$$

This codifies the "disjointification trick" from probability theory and measure theory. Given the connection between differentiation and integration, I guess it makes sense to think of $\mathop{\mathrm{disj}}$ as playing the role of differentiation in the world of boolean algebra. We have a kind of fundamental theorem of calculus, namely:

$$ \mathop{\vec{\bigcup}}_{i \in \mathbb{N}} \mathop{\mathrm{disj}}_{i \in \mathbb{N}} A_i = A_i,$$

and it seems to be the case that $\mathop{\mathrm{disj}}_{i \in \mathbb{N}} A_i$ produces the "smallest" (in the sense of $\subseteq$) sequence making the above formula true; I guess they're order-theoretic adjoints or something.

Questions.

Q0.Is there accepted terminology for what I'm calling "serialization"?

Q1.Is there any existent theory surrounding this concept?

reallychange the notation here, is the operator aplied to the whole family of sets? or is it just applied to the sets whose index is lower than some fixed $i$ ? It makes it unnecessarily more difficult to understand. $\endgroup$ – Max Jul 2 '17 at 15:26