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

Abstract

We propose a new class of mathematical structures called $(m,n)$-semirings (which generalize the usual semirings) and describe their basic properties. We define partial ordering and generalize the concepts of congruence, homomorphism, and so forth, for $(m,n)$-semirings. Following earlier work by Rao (2008), we consider systems made up of several components whose failures may cause them to fail and represent the set of such systems algebraically as an $(m,n)$-semiring. Based on the characteristics of these components, we present a formalism to compare the fault-tolerance behavior of two systems using our framework of a partially ordered $(m,n)$-semiring.