TITLE: Question about definition of Semi algebra
QUESTION [10 upvotes]: 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 

$\emptyset \in S$
If $P_1$, $P_2 \in S$, then $P_1 \cap P_2 \in S$  
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.

REPLY [7 votes]: Here is a counterexample showing that 1,2, and 3 do not prove 3'.
Let $X$ be the nodes of an infinite complete binary tree.  Then for $x\in X$, let $L(x)$ denote all nodes in the left subtree from $x$, and similarly let $R(x)$ denote all nodes in the right subtree from $x$.  Then let
$S = \{\{x\}| x\in X\} \cup \{\{x\}\cup L(x)| x\in X\} \cup \{\{x\}\cup R(x)| x\in X\} \cup \{\{\} \}$
In other words, S is comprised of all singletons, all singletons with their left subtrees, and all singletons with their right subtrees.  One can check that this is a semi-algebra in the sense of 1,2,and 3.  But we will never be able to write $X$ (the complement of the empty set) as a finite disjoint union of elements of $S$.