Finitely Additive, Modular and Probability Functions on Pre-semirings

In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. We also generalize some classical results in probability theory such as the Law of Total Probability, Bayes' Theorem, the Equality of Parallel Systems, and Poincar\'{e}'s Inclusion-Exclusion Theorem. While we prove that modular functions over a couple of known semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring $(Id(D), + ,\cdot)$, where $D$ is a Dedekind domain.