(m,n)-Semirings and a Generalized Fault Tolerance Algebra of Systems

TL;DR

This paper introduces (m,n)-semirings, a new algebraic structure, to model and compare the fault tolerance of complex systems with multiple components, generalizing traditional semiring concepts.

Contribution

The paper defines (m,n)-semirings, extends algebraic concepts to them, and applies this framework to formalize and compare system fault tolerance.

Findings

Defined (m,n)-semirings and their properties

Extended algebraic concepts like congruence and homomorphism

Presented a formalism for comparing system fault tolerance

Abstract

We propose a new class of mathematical structures called (m,n)-semirings} (which generalize the usual semirings), and describe their basic properties. We also define partial ordering, and generalize the concepts of congruence, homomorphism, ideals, etc., for (m,n)-semirings. Following earlier work by Rao, we consider a system as made up of several components whose failures may cause it to fail, and represent the set of systems algebraically as an (m,n)-semiring. Based on the characteristics of these components we present a formalism to compare the fault tolerance behaviour of two systems using our framework of a partially ordered (m,n)-semiring.