There are some set systems with algebraic titles, such as "field", "algebra", "ring" and "semi-ring" (and possibly other titles), in their names. Examples are
- a sigma field (aka sigma algebra, delta algebra),
- a delta ring of sets,
- a sigma ring of sets,
- a field (aka algebra) of sets,
- a ring of sets in order theory sense,
- a ring of sets in measure theory sense,
- a semi-ring of sets,
- a semi-algebra of sets,
among others (I don't know yet but you are welcome to add more).
They seem to suggest some algebraic structures, but it is not the actual algebraic structure at least in one case "a field of sets is not an "field" in the sense of abstract algebra, but a Boolean algebra" (I am not very sure about other cases).
I was wondering if there are some definitions for "field", "algebra", "ring" and "semi-ring" appearing in names of set systems? If not, what are the reasons to name such a set system with one of these titles, instead of randomly pick one?
Why are there some set systems without these algebraic titles in them, such as
- convexity structure,
- $\lambda$ system,
- monotone class,
- $\pi$ system
- closure system?
For example, there is only one set operation finite intersection in defining a $\pi$ system, and only arbitrary intersection in a closure system. So in the spirit of "field" and "ring" for two set operations, shall a $\pi$ system and a closure system be called "group"?
Thanks and regards!