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.