Commonly used algebraic structures over probability, as seen in, for example,
- the convolution semigroup, used in divisible processes (what do you call the semigroup of maximum processes?)
- the general transition semigroup of Markov processes.
- George Lowther introduces this and a connection to quantum probablility in characteristically plain style 1, 2
Applebaum, David. 2009. Lévy Processes and Stochastic Calculus. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.
Ressel, Paul. 1991. “Semigroups in Probability Theory.” In Probability Measures on Groups X, edited by Herbert Heyer, 337–63. Springer US. https://doi.org/10.1007/978-1-4899-2364-6_26.
———. 2011. “A Revision of Kimberling’s Results — with an Application to Max-Infinite Divisibility of Some Archimedean Copulas.” Statistics & Probability Letters 81 (2): 207–11. https://doi.org/10.1016/j.spl.2010.11.008.
Ruzsa, Imre, and Gábor J Székely. 1988. Algebraic Probability Theory. John Wiley & Sons Inc.