{"product_id":"m-n-semirings-and-a-generalized-fault-tolerance-algebra-of-systems-von-syed-eqbal-alam","title":"(m, n)-Semirings and a Generalized Fault Tolerance Algebra of Systems","description":"\u003cp\u003eWe propose a new class of mathematical structures called (m, n)-semirings (which  generalize the usual semirings), and describe their basic properties. We also de¿ne  partial ordering, and generalize the concepts of congruence, homomorphism, etc.,  for (m, n)-semirings. Following earlier work by Rao [37], 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 behavior of  two systems using our framework of a partially ordered (m, n)-semiring. Assuming  0 as a system which is ¿always up¿ and 1 as a system which is ¿always down¿,  based on these assumptions we prove which system is more fault tolerant. We also  compare the fault tolerance behavior of two congruent system using the congruence  relation. We check whether two systems are congruent or not based on their fault  tolerance behavior. We have also mentioned an example of wireless sensor network  and comparison of fault tolerance in two such networks.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783848414192\"\u003e\u003ch3\u003eGeneralization of fault tolerance algebra of systems\u003c\/h3\u003e\u003c\/div\u003e","brand":"Autorenwelt Shop","offers":[{"title":"Softcover - 9783848414192","offer_id":39491112796253,"sku":"9783848414192","price":49.0,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/134c6ef6-d75f-4547-9c37-99084c2ae955.jpg?v=1757743127","url":"https:\/\/shop.autorenwelt.de\/products\/m-n-semirings-and-a-generalized-fault-tolerance-algebra-of-systems-von-syed-eqbal-alam","provider":"Autorenwelt Shop","version":"1.0","type":"link"}