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  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!

Community
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Tim
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1 Answers1

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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.

rschwieb
  • 153,510
  • Thanks! Why are some other set systems not named with these algebraic titles? See my second part. – Tim Jan 07 '13 at 15:07
  • @Tim I'm not sure I have a good feel for what you're asking in the second part, but I'm going to do my best to add something to my solution about your example. The main problem is that I'm not familiar with many of the terms you mentioned. – rschwieb Jan 07 '13 at 15:15
  • Thanks! A field of sets is a canonical complemented distributive lattice, a order-theorectical ring of sets is a canonical distributive lattice. In the same spirit, what kind of algebraic structure (lattice) is a $\pi$ system? – Tim Jan 07 '13 at 18:29
  • @tim It isn't a lattice (it only has one operation). – rschwieb Jan 07 '13 at 20:38
  • It isn't a lattice but can it be something else in some similar spirit? ring/field as to group, as lattice to? – Tim Jan 07 '13 at 20:38
  • @tim I don't know if you noticed, but I added a comment on this at the end of my answer. – rschwieb Jan 07 '13 at 20:41
  • Yes, I noticed it. But it isn't really a semigroup, is it? Or you mean the best analogue, which is the best close algebraic structure although it is not the real one – Tim Jan 07 '13 at 20:42
  • @tim You will have to let me know what semigroup axiom you think fails. EDIT: the "best analogue" is just a subjective opinion. I don't mean to say that there is a hard-and-fast rule. – rschwieb Jan 07 '13 at 20:44
  • A semigroup is defined to satisfy associative property. So I think it is an abelian semigroup. I was wrong. – Tim Jan 07 '13 at 20:46
  • Could you describe what these two binary operations are on fields of sets? I thought we describe them with complements (unary) and finite unions (binary) what is the second binary? – random_0620 Oct 08 '22 at 15:29
  • @SamKirkiles If you look at the wiki you can see the order-theoretic version is "closed under pairwise intersection and union" and that the measure theoretic version's definition also satisfies that. If one wanted to make an analogy with the measure theoretic version maybe they would talk about a ring with involution. – rschwieb Oct 09 '22 at 12:32
  • @rschwieb Isn't symmetric what is difference required to make an addition law on a ring? – random_0620 Oct 09 '22 at 16:29
  • @SamKirkiles I don't understand what you are asking. – rschwieb Oct 09 '22 at 18:34