My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every algebra, there is a unique partition, but isn't this wrong? I mean, in the last example, we can choose the partition $\{\{\omega_1,\omega_2\},\{\omega_1,\omega_2\},\{\omega_3,\omega_4\},\{\omega_5,\omega_6\},\{\omega_7,\omega_8\}\}$, or we can choose $\{\{\omega_1,\omega_2,\omega_3,\omega_4\},\{\omega_5,\omega_6,\omega_7,\omega_8\}\}$?, so the partition is not unique?
Isn't it correct to say that for each partition there is a unique algebra, but for each algebra, there may be many partitions, so you have MORE information if you know the partition, and not just the algebra?
I am not sure, what you refer to as the last example, but there is indeed 1-to-1 correspondence between algebras and partitions at over finite sets. The elements of partitions are atoms of the algebra, that is sets that don't contain any proper non-empty subset which is also an element of algebra. Note that partition $$ E := \{\{\omega_1,\omega_2\},\{\omega_1,\omega_2\},\{\omega_3,\omega_4\},\{\omega_5,\omega_6\},\{\omega_7,\omega_8\}\} $$ does generate the algebra $A$ such that $\{\omega_1,\omega_2\}\in A$, whereas the partition $$ E':=\{\{\omega_1,\omega_2,\omega_3,\omega_4\},\{\omega_5,\omega_6,\omega_7,\omega_8\}\} $$ generates the algebra $A'$ such that $\{\omega_1,\omega_2\}\notin A'$. Perhaps, you are confusing $\{\omega_1,\omega_2\}\in A$ on the level of elements $\omega_i$ rather than sets?
The information flow is rather natural to model with partitions themselvels. I guess, in the stochastic environment you start dealing with algebras and $\sigma$-algebras (which both are a bit more complex than partitions) instead of partitions due to the fact that you'll need to have some measurability properties. In essence, you can consider measurability as information + being nice enough for integration purposes.
Every algebra is a family of sets. It may contain some subfamilies that are partitions. Since algebra contains $\Omega$ by definition, the subfamily $\{\Omega\}$ is of course a partition contained in any algebra. Nevertheless, when we are relating algebras and partitions, we are not talking about partitions that are contained in a given algebra. We are talking about partitions that generate a given algebra. Over finite $\Omega$, for any given algebra $A$ is always a unique partition that generates $A$. That's how you shall understand the 1-to-1 correspondence between algebras and partitions.
The very same statement does not hold for infinite $\Omega$. That is, every partition generates some algebra, but for a given algebra there may not exist a partition that generates it. For example, for any infinite $\Omega$ its powerset is an algebra, and it contains all singletons, i.e. $\{\omega\}\in 2^\Omega$ for all $\omega\in \Omega$. Thus, the only possible partition that could generate the powerset is exactly the collection of all singletons. However, any proper infinite subset of $\Omega$ cannot be obtained as a finite union of singletons, so the powerset of an infinite set does not have a partition it is generated by.