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I found two definitions in some textbooks. One is defined as follows:

A collection $\mathscr{C}$ of subset of $X$ is called a semi-algebra if

(a) $\emptyset \in \mathscr{C},$

(b) if $A, B\in \mathscr{C},$ then $A\cap B\in \mathscr{C},$

(c) for $A\in \mathscr{C},$ there exist finitely many (possibly empty) disjoint sets $E_1, \ldots, E_n\in \mathscr{C}$ such that $ X\setminus A=\bigcup_{i=1}^n E_i.$

Another definition is given by:

A collection $\mathscr{C}$ of subset of $X$ is called a semi-algebra if

(a) $\emptyset \in \mathscr{C},$

(b) if $A, B\in \mathscr{C},$ then $A\cap B\in \mathscr{C},$

(c') For $A, B\in \mathscr{C}$, there exist finitely many disjoint sets $E_1,E_2,\ldots, E_m\in \mathscr{C}$ such that $A\setminus B=\bigcup_{k=1}^m E_k.$

My question is : Are the two definitions equivalent? How to prove that $(c')$ implies $(c)$? The main difficult is that $X$ is not necessary in $\mathscr{C}$.

ljjpfx
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1 Answers1

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The two conditions (call them $(S1)$ and $(S2)$) are not equivalent. $(S1)\Rightarrow (S2)$ is clearly true, but here are a couple of counterexamples that showcase the main obstructions to $(S2)\Rightarrow (S1)$, or at least what I believe them to be:

  • $X=\Bbb R$ and $\mathscr C=\mathcal P(\Bbb R\setminus\{0\})$

  • $X$ an infinite set and $\mathscr C$ the family of finite subsets of $X$.

If you substitute $(a)\equiv[\emptyset\in \mathscr C]$ with $(a')\equiv[X,\emptyset\in \mathscr C]$ they are equivalent. I think that if you substitute $(a)$ with just $(a'')\equiv [X\in\mathscr C]$, then you obtain a condition which is not equivalent to $(a)\land (b)\land (c)$, but which should work just fine with the things you'll need this for.

  • Thank you for your answer. I also found that (a) is replaced by $\emptyset, X\in \mathscr{X}$ in other definition. It seems that it is difficult to search for the original definition. – ljjpfx May 14 '20 at 08:30