Finitely additive, modular, and probability functions on pre-Semirings

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 Poincaré’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),+,⋅), where D is a Dedekind domain.

KEYWORDS:

• Finitely additive functions
• hemiring
• modular functions
• pre-semiring
• probability functions
• semiring

2010 Mathematics Subject Classification:

• 16Y60
• 20M14

Acknowledgements

The first named author is supported by Department of Engineering Science at Golpayegan University of Technology. The authors wish to thank Sepanta Asadi for informing them about a point related to Proposition 4.7. They are also grateful for the useful comments of the anonymous referee.