#### 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.

#### 1. Introduction

Fault tolerance is the property of a system to be functional even if some of its components fail. It is a very critical issue in the design of the systems as in Air Traffic Control Systems [1, 2], real-time embedded systems [3], robotics [4, 5], automation systems [6, 7], medical systems [8], mission critical systems [9], and a lot of others. Description of fault-tolerance modeling using algebraic structures is proposed by Beckmann [10] for groups and by Hadjicostis [11] for semigroups and semirings. Semirings are also used in other areas of computer science like cryptography [12], databases [13], graph theory, game theory [14], and so forth. Rao [15] uses the formalism of semirings to analyze the fault tolerance of a system as a function of its composition, with a partial ordering relation between systems used to compare their fault-tolerance behaviors.

The generalization of algebraic structures was in active research for a long time; Timm [16] in 1967 proposed commutative -groups; later Crombez [17] in 1972 generalized rings and named it as -rings. It was further studied by Crombez and Timm [18], Leeson and Butson [19, 20], and by Dudek [21]. Recently the generalization of algebraic structures is studied Davvaz et al. [22, 23].

In this paper, we first define the -semiring (which is a generalization of the ordinary semiring , where is a set with binary operations + and ), using and which are -ary and -ary operations, respectively. We propose identity elements, multiplicatively absorbing elements, idempotents, and homomorphisms for -semirings. We also briefly touch on zero-divisor free, zero-sum free, additively cancellative, and multiplicatively cancellative -semirings and the congruence relation on -semirings. In Section 4, we use the facts that each system consists of components or subsystems and that the fault-tolerance behavior of the system depends on each of the components or subsystems that constitute the system. A system may itself be a module or part of a larger system, so that its fault tolerance affects that of the whole system of which it is a part. We analyze the fault tolerance of a system given its composition, extending earlier work of Rao [15]. Section 2 describes the notations used and the general conventions followed.

Section 3 deals with the definition and properties of -semirings. In Section 4, we extend the results of Rao [15] using a partial ordering on the -semiring of systems: the class of systems is algebraically represented by an -semiring, and the fault-tolerance behavior of two systems is compared using partially ordered -semiring.

#### 2. Preliminaries

The set of integers is denoted by , with and denoting the sets of positive integers and negative integers, respectively, and and used are positive integers. Let be a set and a mapping ; that is, is an -ary operation. Elements of the set are denoted by where .

*Definition 1. *A nonempty set with an -ary operation is called an *-ary groupoid* and is denoted by (see Dudek [24]).

We use the following general convention.

The sequence is denoted by where .

For all , the following term is represented as

In the case when , (2) is expressed as

*Definition 2. *Let be elements of set .(i) Then, the associativity and distributivity laws for the -ary operation are defined as follows.(a) *Associativity*:
?for all , for all (from Gluskin [25]).(b) *Commutativity*:
?for every permutation of (from Timm [16]), . (ii) An -ary groupoid is called an *-ary semigroup* if is associative (from Dudek [24]); that is, if
?for all , where .(iii) Let , be elements of set , and . The -ary operation is *distributive* with respect to the -ary operation if

*Remark 3. *(i) An -ary semigroup is called a *semiabelian* or -commutative if
for all (from Dudek and Mukhin [26]).

(ii) Consider a -ary group in which the -ary operation is distributive with respect to itself, that is,
for all . These types of groups are called *autodistributive *-ary groups (see Dudek [27]).

#### 3. -Semirings and Their Properties

*Definition 4. *An -semiring is an algebraic structure which satisfies the following axioms:(i) is an -ary semigroup,(ii) is an -ary semigroup,(iii)the -ary operation is *distributive* with respect to the -ary operation .

*Example 5. *Let be any Boolean algebra. Then, is an -semiring where and , for all and .

In general, we have the following.

Theorem 6. *Let be an ordinary semiring. Let be an -ary operation and be an -ary operation on as follows:
**
Then, is an -semiring.*

*Proof. *Omitted as obvious.

*Example 7. *The following give us some -semirings in different ways indicated by Theorem 6.(i) Let be an ordinary semiring and be in . If we set
?we get a -semiring .(ii) In an -semiring , fixing elements and , we obtain two binary operations as follows:
?Obviously, is a semiring.(iii) The set of all negative integers is not closed under the binary products; that is, does not form a semiring, but it is a -semiring.

*Definition 8. *Let be an -semiring. Then -ary semigroup has an *identity element * if
for all and . We call as an *identity element* of -semiring .

Similarly, -ary semigroup has an *identity element * if
for all and .

We call as an *identity element* of -semiring .

We therefore call the -identity, and the -identity.

*Remark 9. *In an -semiring , placing and , and times, respectively, we obtain the following binary operations:

*Definition 10. *Let be an -semiring with an -identity element and -identity element . Then,(i) is said to be *multiplicatively absorbing* if it is absorbing in , that is, if
?for all .(ii) is called *zero-divisor free* if
?always implies or or or .?Elements are called *left zero-divisors* of -semiring if there exists and the following holds:
(iii) is called *zero-sum free* if
?always implies .(iv) is called *additively cancellative* if the -ary semigroup is cancellative, that is,
?for all and for all .(v) is called *multiplicatively cancellative* if the -ary semigroup is cancellative, that is,
?for all and for all . ?Elements are called *left cancellable* in an -ary semigroup if
?for all .? is called *multiplicatively left cancellative* if elements are multiplicatively left cancellable in -ary semigroup .

Theorem 11. *Let be an -semiring with -identity .*(i)*If elements are multiplicatively left cancellable, then elements ,,, are not left divisors.*(ii)*If the -semiring is multiplicatively left cancellative, then it is zero-divisor free.*

We have generalized Theorem 11 from Theorem 4.4 of Hebisch and Weinert [28].

We have generalized the definition of idempotents of semirings given by Bourne [29] and Hebisch and Weinert [28]), as follows.

*Definition 12. *Let be an -semiring. Then,(i) it is called *additively idempotent* if is an idempotent -ary semigroup, that is, if
?for all ;(ii) it is called *multiplicatively idempotent* if is an idempotent -ary semigroup, that is, if
?for all , .

Theorem 13. *An -semiring having at least two multiplicatively idempotent elements in the center is not multiplicatively cancellative.*

*Proof. *Let and be two multiplicatively idempotent elements in the center, . Then,
which can be written as follows:
which is represented as

If the -semiring is multiplicatively cancellative, then the following holds true:
which implies that , which is a contradiction to the assumption that ; therefore, is not multiplicatively cancellative.

We have generalized Exercise 2.7 in Chapter of Hebisch and Weinert [28] to get the following.

*Definition 14. *Let ( be an -semiring and an equivalence relation on . (i) Then, is called a *congruence relation* or a *congruence* of , if it satisfies the following properties for all and : (a) if then ,(b) if then ,?for all , , , .(ii) Let be a congruence on an algebra . Then, the *quotient* of by , written as , is the algebra whose universe is and whose fundamental operation satisfies
?where [30].

Theorem 15. *Let be an -semiring and the relation be a congruence relation on . Then, the quotient is an -semiring under and , for all and in .*

*Proof. *Omitted as obvious.

*Definition 16. *We define homomorphism, isomorphism, and a product of two mappings as follows.(i) A mapping from -semiring into -semiring is called a *homomorphism* if
?for all , .(ii)The -semirings and are called *isomorphic* if there exists one-to-one homomorphism from onto . One-to-one homomorphism is called *isomorphism*.(iii) If we apply mapping and then on , we get the mapping which is equal to , where . It is called the *product* of and [28].

We have generalized Definition 16 from Definition 2 of Allen [31].

We have generalized the following theorem from Theorem given by Hebisch and Weinert [28].

Theorem 17. *Let , , and be -semirings. Then, if the following mappings and*?*? are homomorphisms; then,*??* is also a homomorphism.*

*Proof. *Let and be in . Then
In a similar manner, we can deduce that
Thus, it is evident that is a homomorphism from .

This proof is similar to that of Theorem 6.5 given by Burris and Sankappanavar [30].

*Definition 18. *Let and be -semirings and a homomorphism. Then, the *kernel* of , written as ker?, is defined as follows:
Generalization of Burris and Sankappanavar [30].

Theorem 19. *Let and be -semirings and a homomorphism. Then, is a congruence relation on , and there exists a unique one-to-one homomorphism from into .*

*Proof. *Omitted as obvious.

Corollary 20. *Let be an -semiring and and congruence relations on , with . Then, is a congruence relation on , and .*

Lemma 21. *Let , . Then,*(i)*, ,*(ii)*, .*

*Proof. *(i)
By associativity (Definition 2 (i)), (34) is equal to
(ii) Similar to part (i).

#### 4. Partial Ordering on Fault Tolerance

In this sections we use , and so forth, where to denote individual system components that are assumed to be *atomic* at the level of discussion; that is, they have no components or subsystems of their own. We use *component* to refer to such an atomic part of a system, and *subsystem* to refer to a part of a system that is not necessarily atomic. We assume that components and subsystems are disjoint, in the sense that if they fail, they fail independently and do not affect the functioning of other components.

Let be a universal set of all systems in the domain of discourse as given by Rao [15], and let be a mapping , that is, is an -ary operation. Likewise, let be an -ary operation.

*Definition 22. *We define and operations for systems as follows.(i) is an -ary operation which applies on systems made up of components or subsystems, where if any one of the components or subsystems fails, then the whole system fails. ?If a system made up of components , then, the system over operation is represented as for all . The system fails when any of the components fails.(ii) is an -ary operation which applies on a system consisting of components or subsystems, which fails if all the components or subsystems fail; otherwise it continues working even if a single component or subsystem is working properly.?Let a system consist of components , then, the system over operation is represented as for all . The system fails when all the components fail.

Consider a partial ordering relation on , such that is a partially ordered set (poset). This is a *fault-tolerance partial ordering* where means that has a lower measure of some fault metric than and has a better fault tolerance than , for all (see Rao [32] for more details) and , are disjoint components.

Assume that represents the atomic system “which is always up” and represents the system “which is always down” (see Rao [32]).

*Observation 23. *We observe the following for all disjoint components , , which are in .(i) for all .?This is so since represents the component or system which never fails, and as per the definition of , the system as a whole fails if all the components fail, and otherwise it continues working even if a single component is working properly. In a system , even if all other components and fail even then is up and the system is always up.(ii) for all .?This is so since represents the component or system which is always down, and as per the definition of if either of the component fails, then the whole system fails. Thus, even though all other components are working properly but due to the component the system is always down.

*Definition 24. *If is an -semiring and is a poset, then is a *partially ordered **-semiring* if the following conditions are satisfied for all , , and , .(i)If , then .(ii)If , then .

*Remark 25. *As it is assumed that is the system which is always up, it is more fault tolerant than any of the other systems or components. Therefore , for all . Similarly, because is the system that always fails, and therefore, it is the least fault tolerant; every other system is more fault tolerant than it.

*Observation 26. *The following are obtained for all disjoint components , , , , , , which are in , where , .(i).(ii).(iii).(iv).

From the above description of and , the observation is quite obvious. Case (i) shows that is less faulty than , and is less faulty than . Similarly, case (ii) shows that is more fault tolerant than and is more fault tolerant than . Likewise, case (iii) shows the operation over , and of to be less faulty than and more faulty than , and a similar interpretation is made for (iv).

Lemma 27. *If is a fault-tolerance partial order and , , , are disjoint components, which are in , where , then for all and the following holds true:*(i)*if , then , *(ii)*if , then .*

*Proof. *(i) Since for all , we have
which is represented as follows:
By operation on both sides of (37) with , we get
By operation on both sides of (38) with
From (39) and (40), we get
Similarly, we find for terms
From Lemma 21, (42) may be represented as
so

(ii) Since , for all
After following similar steps as seen in part (i), we use the operation for terms,
which is represented as
and so

Theorem 28. *If is a fault-tolerance partial order and given disjoint components , , , in , where , and , the following obtain.*(i)*If , then
*(ii)*If , then
*

*Proof. *(i) Since , for all .

Therefore, from Lemma 27 (i)
From Definition 24 of a partially ordered -semiring, we deduce that
for all .

(ii) Since , for all , from Lemma 27 (ii), we find that
From Definition 24 of a partially ordered -semiring, we deduce that
for all .

Lemma 29. *If?? is a fault-tolerance partial order and are disjoint components which are in , where and , one gets the following:*(i)*,*(ii)*.*

*Proof. * (i) As
by operation on both sides of (55) with , we get
Therefore,
Similarly, we obtain
Hence,
for all .

(ii) As
by operation on both sides of (60) with , we get
Similarly, we obtain
Hence,
for all .

Corollary 30. *If is a fault-tolerance partial order, then the following hold for all disjoint components which are elements of , where , and :*(i)*, where ,*(ii)*, where .*

*Proof. *(i) From (58), we deduce that
Therefore,

(ii) As in part (i), we deduce from (62) that

represents the system which is obtained after applying the operation on repeated systems or subsystems. Similarly, represents the system which is obtained after applying the operation on repeated systems or subsystems.

Theorem 31. *If is a fault-tolerance partial order, and components , are disjoint components and are in , then*(i)*,*(ii)*.*

Corollary 32. *The following hold for all disjoint components , , , , which are elements of , where .*(i)*If , then
*(ii)*If , then
*

*Proof. *(i) and from Theorem 31, . Therefore, .

(ii) The proof is very similar to that of part (i).

Corollary 33. *Let and be positive integers and , . Given disjoint components , , , that are in , the following hold:*(i)*If , then .*(ii)*If , then .*

*Proof. *Similar to Corollary 32.

Theorem 34. *Let be a fault-tolerance partial order and and for all , where and . Then, the following obtain:*(i)*,*(ii)*,*(iii)*,*(iv)*.*

*Proof. * (i) As
from Lemma 27 (i), we get
This is written as
So by operation on both sides of (71) with , we get
So by operation on both sides of (71) with , we get
From (72) and (73), we get
Similarly, we get for terms

(ii) We know that
From Lemma 27 (ii), we get
Which is represented as follows
Now by operation on both sides of (78) with , we get
So by operation on both sides of (78) with , we get
So now from (79) and (80), we get
Similarly, we find for terms

(iii) From Lemma 27 (ii)
Similar to part (i), we find operation of terms and get

(iv) We know that
so from Lemma 27 (i), we get
As proved in part (ii), we find operations of terms and get
Thus, we get

Corollary 35. *If?? is a fault-tolerance partial order and , where , if , for all disjoint components , , , , which are in , where and , then*(i)*,
*(ii)*.*

*Proof. *(i) Proof is similar to that of Theorem 34 (i). We find the operation of terms where , and .

(ii) Proof is similar to that of Theorem 34 (ii). We find the operation of terms where , and .

We propose the following theorem for very complex systems.

Theorem 36. *If is a fault-tolerance partial order, disjoint components , , , , , , , are in and , , and , where , , and , then*(i)*, for all ; *(ii)