Is it easy to see if $\alpha $ is an algebra or a semiring? I have $\alpha \subset P(\Omega) $ ( $P(\Omega)$ powerset of $\Omega$ ) where $ A, B\in \alpha \Rightarrow A\cup B \in \alpha, A \cap B \in \alpha$. Is $\alpha$ an algebra or a semiring?
I know that if $\Omega \neq \varnothing $ and $\alpha \subset P(\Omega) $ then $\alpha -$algebra $\Leftrightarrow $
i) $\Omega \in \alpha$ (or $\Omega , \varnothing \in \alpha$)
ii) $A \in \alpha \Rightarrow A^{c} \in \alpha $
iii) $A \in \alpha \Rightarrow A \cup B \in \alpha (\Leftarrow) A \cap B \in \alpha$
(or if $A_{i} \in \alpha ,i=1,2,...,n \Rightarrow \bigcup_{i=1}^{n}A_{i} \in \alpha$)
I know that $\alpha \subset P(\Omega) $ is called a semiring if:
i) $A, B \in \alpha \Rightarrow A \cap B \in \alpha$
ii) $A, B \in \alpha \Rightarrow$ there is a finite family $\begin{Bmatrix} A_{j}, j=1,2,...,n \end{Bmatrix}$ with $A_{j} \in \alpha$ for any $j$ and $A_{j} \cap A_{k} = \varnothing$ for any $ j \neq k$ :
$A-B = \bigcup_{j=1}^{n}A_{j}$
So, I don't have enough information to say that $\alpha$ is an algebra but do I have enough to say that $\alpha$ is a semiring?
Let $\alpha = \{ \Omega \} $.
$\Omega, \Omega \in \alpha \Rightarrow \Omega \cup \Omega = \Omega \in \alpha$ and
$\Omega \cap \Omega = \Omega \in \alpha$ .
But $\varnothing = \Omega^{c} \notin \alpha$
So $\alpha$ is neither an algebra nor a semiring