Given a semi-ring of sets $S$, the by $S$ generated ring of sets $R$ is the set of all sets $A$ which are unions of pairwise disjoint sets $C_i\in S$ ($\bigsqcup$ shall be the disjoint union):
$$A\in R \iff \exists C_1,\ldots,C_n\in S: A = \bigsqcup_{i=1}^n C_i$$
If I extend this definition to countable unions, what type of set structure would I get? Let $T$ be a family of sets where all sets are countable unions of pairwise disjoint sets from the semi-ring:
$$A\in T \iff \exists C_i\in S: A = \bigsqcup_{i=1}^\infty C_i$$
What is the term for the set structure $T$? Is $T$ the same as the $\sigma$-algebra generated by $S$? If not: How is such a structure called?
My motivation: The set of half-open intervals (rectangles, cuboids, ...) is a semi-ring. I want to know how to call the family of sets which are countable disjoint unions of half-open intervals (rectangles, cuboids, ...).
The term you're looking for is $\sigma$-ring. The difference between a $\sigma$-ring and a $\sigma$-algebra is the requirement that $\sigma$-algebras be closed under complements.