Finitely Additive, Modular and Probability Functions on Pre-semirings

TL;DR

This paper introduces finitely additive, probability, and modular functions on semiring-like structures, extending classical probability results and exploring their properties over various algebraic systems.

Contribution

It generalizes classical probability theorems to semiring contexts and characterizes modular functions over different semirings, including Dedekind domains.

Findings

Finitely additive functions relate to complemented elements in semirings.

Classical probability laws are generalized within semiring frameworks.

Modular functions over certain semirings can be highly varied, not necessarily constant.

Abstract

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.