Making sense out of "field", "algebra", "ring" and "semi-ring" in names of set systems

1. 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?

2. Why are there some set systems without these algebraic titles in them, such as

   • topology,
   • 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!

Always remember that the names we pick for things are a matter of convenience, and there are not really any rules to follow. (But it helps when people pick predictable things!)

Here's a fast answer that is probably not historically accurate, but will probably put your mind at ease: in universal algebra an algebra is just some set with different operations and rules acting in it. In that sense, groups, rings, rings of sets etc are all just generic "algebras". So you can see some people (at least) don't mind using "algebra" very flexibly.

As you have found out, the -set versions of rings and algebras are a little different from the algebraic ones. Let's focus on the similarities for a moment, to see why the names are kind of parallel to each other:

1. Ring and ring-of-sets: Both involve a set closed under two operations.

2. Field and field-of-sets: Both involve a set closed under two operations, plus a unary operation (multiplicative inverse/complementation)

The case of a "Boolean algebra" is interesting, because it kind of lies at the intersection of these two notions. While someone said they are lattice theoretic, it is also important to remember that they really are honest-to-goodness rings, too.

The use of "semi-" in front of terms has a pretty consistent use, and that is just to say that it is not quite as strong as the usual version. This is true for both a semi-ring-of-sets and a semiring.

To find an analogue of $\sigma$-algebras in ring theory, we would have to think of a field with infinitary operations; however, I don't know if anything like that exists. I do have an easy example of a semiring with infinitary operations, and that is the semiring of ideals of a ring. (That is, the set is the set of ideals of a fixed ring, along with the operations of ideal addition and ideal multiplication.)

For your example of a $\pi$-system, I think the best analogue is a semigroup, since there are no "inverses" provided by the complement. If you took a $\pi$-system and required it to be closed under complements, then I would be more inclined to analogize that to a group.