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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 482391, 10 pages
http://dx.doi.org/10.1155/2013/482391
Research Article

-Semirings and a Generalized Fault-Tolerance Algebra of Systems

1College of Computers and Information Technology, Taif University, Taif 21974, Saudi Arabia
2IIIT-Bangalore, Bangalore 560 100, India
3Department of Mathematics, Yazd University, Yazd, Iran

Received 21 July 2012; Revised 31 December 2012; Accepted 31 December 2012

Academic Editor: Ray K. L. Su

Copyright © 2013 Syed Eqbal Alam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose a new class of mathematical structures called -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 -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 -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 -semiring.