Template talk:Families of sets

Semialgebra

I recently edited https://en.wiki.x.io/wiki/Semiring#Semiring_of_sets and https://en.wiki.x.io/wiki/Ring_of_sets#Related_structures to change the erroneous definition of a semialgebra. It previously said:

• A semialgebra on $X$ is a semiring that has $X$ as an element.

I now have it as a semiring ${\mathcal {S}}$ with the extra condition:

• If $E\in {\mathcal {S}}$ then there exists a finite number of mutually disjoint sets $C_{1},\ldots ,C_{n}\in {\mathcal {S}}$ such that $X\setminus E=\bigcup _{i=1}^{n}C_{i}.$

This is consistent with virtually all sources I can find, including (oddly enough) the source it originally cited (Durrett 2019). Note that these are not equivalent: the first implies the second, but the converse fails for ${\mathcal {S}}:=\{\emptyset ,\{x\},\{y\}\}$ on $X=\{x,y\}$. Besides that, the second one is a natural starting point for the Caratheodory construction, but the extra condition of having $X$ in your collection doesn't give you anything there (that I know of).

Anyhow, I just changed that part of the template to reflect this. Unfortunately, it make "semialgebra" identical to "semiring" from the perspective of the table. Given that the distinction between them is rather subtle and arcane outside of measure theory, I guess this isn't surprising.

- Morgan 129.72.100.78 (talk) 04:37, 6 October 2022 (UTC)