8
$\begingroup$

I am wondering if someone could help me with basic properties of semi algebra. We say that $S$ is a semi algebra of subsets of X if

  1. $\emptyset \in S$
  2. If $P_1$, $P_2 \in S$, then $P_1 \cap P_2 \in S$
  3. If $P \in S$, then $X \backslash P$ can be written as a finite union of sets from $S$.

But I am finding that sometimes it is defined using the following 3' instead of 3.

3'. If $P \in S$, then $X \backslash P$ can be written as a disjoint finite union of sets from $S$.

My question is are these definitions equivalent? If so can someone please show me how we can obtain 3' from the first three conditions?

Thank you.

  • 0
    I am used to seeing semi-algebra defined with 2,3', without the first condition.2013-06-29
  • 0
    This thread was confusing because a seemingly incorrect answer has been accepted. Other answers don't seem fully confident that their answers are correct. I would like a correct answer. I have therefore reposted this question here http://math.stackexchange.com/questions/1135203/question-about-the-definition-of-a-semialgebra Future readers or answers should look to this new thread if they find this one unhelpful. Hopefully this new thread can come to an accepted answer to this question that is correct this time.2015-02-06

2 Answers 2