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While studying some measure theory, I came across the following two definitions, given a set $\mathbb{X}$:

  1. A ring is a non-empty subset of $\mathcal{P}(\mathbb{X})$ such that it is closed for unions and set-differences
  2. An algebra is a non-empty subset of $\mathcal{P}(\mathbb{X})$ such that it is a ring and contains $\mathbb{X}$; equivalently, it is closed for unions and complements

This immediately got me thinking about the algebraic implications. Given a former question of mine on this site, I saw that the first definition actually said such a set was a subring of $\mathcal{P}(\mathbb{X})$, with the operations $A \Delta B = (A\setminus B) \cup (B \setminus A)$ and $A\cup B$.

But this left me with the second one: algebras, as far as I've seen, are defined over fields. So what field is that algebra talking about? If there isn't any, why is the term "algebra" used?

As a side note, is there an Algebraic equivalent of "$\sigma$-algebras" and "$\sigma$-rings", without talking about the particular context of subrings of $\mathcal{P}(\mathbb{X})$?

Thanks in advance!

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    $\begingroup$ I don't know the history of these terms off-hand (and don't want to spend time now flipping through some of my books looking for it), but it's worth mentioning that math terms didn't all come into existence at the same time. Often certain terms were used for many years before other uses of the term showed up, so asking why "algebra" is used while insisting on a particular current usage of the word for something defined over a field is a bit short-sighted. As for your side note, you'll want to look into the subject of Boolean rings and Boolean algebras. $\endgroup$
    – Dave L. Renfro
    Commented Aug 17, 2021 at 20:01
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    $\begingroup$ (finished what I had been working on) Regarding your side note, see Representation of sigma complete Boolean algebras and Complete Boolean algebra not isomorphic to a $\sigma$-algebra. As for "algebra", the basic relations involving unions and intersections and such of sets had been, in the early 1900s (and perhaps earlier), referred to as algebraic operations on sets, so at some point instead of fields of sets (term still in use), I suppose "algebras of sets" came be be used. $\endgroup$
    – Dave L. Renfro
    Commented Aug 17, 2021 at 21:12
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    $\begingroup$ I did some looking around just now, and although I looked in books a lot less well known than Halmos' 1950 "Measure Theory", it was the Halmos book (References, p. 291, for Section 4: Rings and algebras -- cites [27] in the Bibliography) that led me to what might be the origin of the terms. Halmos' [27] is the 1927 2nd edition of Hausdorff's set theory book. I don't know to what extent these terms (German equivalents, I suppose) are in the 1914 1st edition of Hausdorff's book, (continued) $\endgroup$
    – Dave L. Renfro
    Commented Aug 18, 2021 at 7:35
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    $\begingroup$ but footnote 1) on p. 77 of the 1927 2nd edition, which I happen to have a copy of, states: "Die Namen Ring und Körper beruhen auf einer schwachen Analogie zu den gleichbenannten Begriffen der Zahlentheorie." Presumably the same wording appears in the 1935 3rd edition (which I don't have a copy of), because the 1957 English translation (Chelsea Publishing Company) of the 3rd edition has the following footnote on p. 90: "The names ring and field are used because of some slight analogy to the similarly-named concepts of Number Theory." (continued) $\endgroup$
    – Dave L. Renfro
    Commented Aug 18, 2021 at 7:42
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    $\begingroup$ See also Were $\sigma$-algebras defined for probability?, including the comments. I don't know whether the use of these terms for collections of sets can be found earlier than 1927, but at least for now 1927 is an upper bound for this usage of these terms, including "ring", which is probably the most significant term here (since as I've said and you already know, "algebra" was long before this used as an adjective for many mathematical descriptions). $\endgroup$
    – Dave L. Renfro
    Commented Aug 18, 2021 at 7:42

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