Commonly used algebraic structures over probability, as seen in, for example, Free probability.
1 Algebraic probability
In algebraic probability, we do not take the Kolmogorov axioms as foundational. Instead, we do away with measure theory and event spaces, starting rather from RVs and expectations.
George Lowther introduces this and a connection to quantum probability in a characteristically plain-talk style 1, 2, which is one useful generalization. We can also get a handle on “non-commutative” probability this way and are especially interested in free probability in that context. But my knowledge is exhausted now. If you wish to know more, here are some people who actually know stuff about it:
- Terry Tao, 254A, Notes 5: Free probability
- Speicher survey articles
- Speicher’s blog Free Probability Theory
- Roland Speicher’s other surveys
2 Group structures which arise in classic probability
- the convolution semigroup, used in divisible processes (what do you call the semigroup of maximum processes?)
- the general transition semigroup of Markov processes.
There is obviously a lot going on. But I do not know it. See, however, John Baez’s category theory lists.