ABSTRACT

We propose a new class of algebraic structure named as (m, n)-semihyperring which is a generalization of usual semihyperring. We define the basic properties of (m, n)-semihyperring like identity elements, weak distributive (m, n)- semihyperring, zero sum free, additively idempotent, hyperideals, homomorphism, inclusion homomorphism, congruence relation, quotient (m, n)-semihyperring etc. We propose some lemmas and theorems on homomorphism, congruence relation, quotient (m, n)-semihyperring, etc. and prove these theorems. We further extend it to introduce the relationship between fuzzy sets and (m, n)-semihyperrings and propose fuzzy hyperideals and homomorphism theorems on fuzzy (m, n)-semihyperrings and the relationship between fuzzy (m, n)-semihyperrings and the usual (m, n)-semihyperrings.

1. Introduction

A semihyperring is essentially a semiring in which addition is a hyperoperation [1]. Semihyperring is in active research for a long time. Vougiouklis [2] generalize the concept of hyperring by dropping the reproduction axiom where and are associative hyper operations and distributes over and named it as semihyperring. Chaopraknoi, Hobuntud and Pianskool [3] studied semihyperring with zero. Davvaz and Poursalavati [4] introduced the matrix representation of polygroups over hyperring and also over semihyperring. Semihyperring and its ideals are studied by Ameri and Hedayati [5].

Zadeh [6] introduced the notion of a fuzzy set that is used to formulate some of the basic concepts of algebra. It is extended to fuzzy hyperstructures, nowadays fuzzy hyperstructure is a fascinating research area. Davvaz introduced the notion of fuzzy subhypergroups in [7], Ameri and Nozari [8] introduced fuzzy regular relations and fuzzy strongly regular relations of fuzzy hyperalgebras and also established a connection between fuzzy hyperalgebras and algebras. Fuzzy subhypergroup is also studied by Cristea [9]. Fuzzy hyperideals of semihyperrings are studied by [1,10,11].

The generalization of Krasner hyperring is introduced by Mirvakili and Davvaz [12] that is named as Krasner (m, n) hyperring. In [13] Davvaz studied the fuzzy hyperideals of the Krasner (m, n)-hyperring. Generalization of hyperstructures are also studied by [1,14-16].

In this paper, we introduce the notion of the generalization of usual semihyperring and called it as (m, n)- semihyperring and set fourth some of its properties, we also introduce fuzzy (m, n)-semihyperring and its basic properties and the relation between fuzzy (m, n)-semihyperring and its associated (m, n)-semihyperring.

The paper is arranged in the following fashion:

Section 2 describes the notations used and the general conventions followed. Section 3 deals with the definitions of (m, n)-semihyperring, weak distributive (m, n)- semihyperring, hyperadditive and multiplicative identity elements, zero, zero sum free, additively idempotent and some examples of (m, n)-semihyperrings.

Section 4 describes the properties of (m, n)-semihyperring. This section deals with the definitions of hyperideals, homomorphism, congruence relation, quotient of (m, n)-semihyperring and also the theorems based on these definitions.

Section 5 deals with the fuzzy (m, n)-semihyperrings, fuzzy hyperideals and homomorphism theorems on (m, n)- semihyperrings and fuzzy (m, n)-semihyperrings.

2. Preliminaries

Let be a non-empty set and be the set of all non-empty subsets of. A hyperoperation on is a map and the couple is called a hypergroupoid. If A and B are non-empty subsets of, then we denote,

and.

Let be a non-empty set, be the set of all nonempty subsets of and a mapping is called an m-ary hyperoperation and m is called the arity of hyperoperation [14].

A hypergroupoid is called a semihypergroup if for all we have which means that

Let f be an m-ary hyperoperation on and subsets of. We define

for all.

Definition 2.1 is a semihyperring which satisfies the following axioms:

1) is a semihypergroup;

2) is a semigroup and;

3) distributes over,

and

for all [3].

Example 2.2 Let be a semiring, we define 

1)

2)

Then is a semihyperring.

An element 0 of a semihyperring is called a zero of if and [3].

The set of integers is denoted by, with and denoting the sets of positive integers and negative integers respectively. Elements of the set are denoted by where.

We use following general convention as followed by [10,17-19]:

The sequence is denoted by.

The following term:

(1)

is represented as:

(2)

In the case when, then (2) is expressed as:

Definition 2.3 A non-empty set with an m-ary hyperoperation is called an m-ary hypergroupoid and is denoted as. An m-ary hypergroupoid is called an m-ary semihypergroup if and only if the following associative axiom holds:

for all and [14].

Definition 2.4 Element e is called identity element of hypergroup if

for all and [14].

Definition 2.5 A non-empty set with an n-ary operation g is called an n-ary groupoid and is denoted by [19].

Definition 2.6 An -ary groupoid is called an n-ary semigroup if g is associative, i.e.,

for all and [19].

3. Definitions and Examples of (m, n)-Semihyperring

Definition 3.1 is an (m, n)-semihyperring which satisfies the following axioms:

1) is a m-ary semihypergroup;

2) is an n-ary semigroup;

3) is distributive over f i.e.,

Remark 3.2 An (m, n)-semihyperring is called weak distributive if it satisfies Definition 3.1 1), 2) and the following:

Remark 3.2 is generalization of [20].

Example 3.3 Let be the set of all integers. Let the binary hyperoperation and an n-ary operation g on which are defined as follows:

and

.

Then is called a -semihyperring.

Example 3.3 is generalization of Example 1 of [1].

Definition 3.4 Let e be the hyper additive identity element of hyperoperation f and be multiplicative identity element of operation g then

for all and and

for all and.

Definition 3.5 An element 0 of an (m, n)-semihyperring is called a zero of if

for all.

for all.

Remark 3.6 Let be an (m, n)-semihyperring and e and be hyper additive identity and multiplicative identity elements respectively, then we can obtain the additive hyper operation and multiplication as follows:

and for all.

Definition 3.7 Let be an (m, n)-semihyperring.

1) (m, n)-semihyperring is called zero sum free if and only if implies .

2) (m, n)-semihyperring is called additively idempotent if be a m-ary semihypergroup, i.e. if.

4. Properties of (m, n)-Semihyperring

Definition 4.1 Let be an (m, n)-semihyperring.

1) An m-ary sub-semihypergroup of is called an (m, n)-sub-semihyperring of if, for all.

2) An m-ary sub-semihypergroup of is called a) a left hyperideal of if, and.

b) a right hyperideal of if, and.

If is both left and right hyperideal then it is called as an hyperideal of.

c) a left hyperideal of an (m, n)-semihyperring of is called weak left hyperideal of if for and then or implies.

Definition 4.1 is generalization of [21].

Proposition 4.2 A left hyperideal of an (m, n)-semihyperring is an (m, n)-sub-semihyperring.

Definition 4.3 Let and be two (m, n)-semihyperrings. The mapping is called a homomorphism if following condition is satisfied for all,.

and

Remark 4.4 Let and be two (m, n)-semihyperrings. The mapping for all, is called an inclusion homomorphism if following relations hold:

and

Remark 4.4 is generalization of [7].

Theorem 4.5 Let, and be (m, n)-semihyperrings. If mappings and are homomorphisms, then is also a homomorphism.

Proof. Omitted as obvious.

Definition 4.6 Let be an equivalence relation on the (m, n)-semihyperring and A_i and B_i be the subsets of for all. We define for all there exists such that holds true and for all there exists such that holds true [22].

An equivalence relation is called a congruence relation on if following hold:

1) for all,; if

then, where and2) for all,; if then, where [23].

Lemma 4.7 Let be an (m, n)-semihyperring and be the congruence relation on then 1) if then

for all

2) if then following holds:

for all

Proof.

1) Given that

(3)

for all. Let e be the hyper additive identity element, then (3) can be represented as follows:

(4)

do f hyperoperation on both sides of (4) with to get

(5)

(6)

(7)

do f hyperoperation on both sides of (7) with to get the following equation:

(8)

(9)

(10)

Similarly we can do f hyperoperation till to get the following result:

(11)

Which can also be represented as:

(12)

2) Given that

(13)

for all. Let be the multiplicative identity element

(14)

do g hyperoperation on both sides of (14) with to get

(15)

(16)

(17)

do g hyperoperation on both sides of (17) with to get the following equation:

(18)

(19)

(20)

Similarly we can do g operation till to get the following result:

Theorem 4.8 Let be an (m, n)-semihyperring and be the congruence relation on. Then if and for all and then the following is obtained: for all

Proof. Can be proved similar to Lemma 4.7.

Definition 4.9 Let be a congruence on. Then the quotient of by, written as, is the algebra whose universe is and whose fundamental operation satisfy

where [23].

Theorem 4.10 Let be an (m, n)-semihyperring and be the equivalence relation and strongly regular on then is also an (m, n)-semihyperring.

Definition 4.11 Let be an (m, n)-semihyperring and be the congruence relation. The natural map is defined by and where for all, .

Theorem 4.12 Let and be two congruence relations on (m, n)-semihyperring such that. Then

is a congruence on and

Proof. Similar to [24], we can deduce that is an equivalence relation on. Suppose for all and for all . Since is congruence on therefore and which implies

and respectively, therefore is a congruence on.

Theorem 4.13 The natural map from an (m, n)-semihyperring to the quotient of the (m, n)-semihyperring is an onto homomorphism.

Definition 4.11 and Theorem 4.13 is generalization of [23].

Proof. let be the congruence relation on (m, n)- semihyperring and the natural map be . For all, where following holds true:

In a similar fashion we can deduce for, for all, where:

So is onto homomorphism.

Proof is similar to [23].

5. Fuzzy (m, n)-Semihyperring

Let be a non-empty set. Then 1) A fuzzy subset of is a function;

2) For a fuzzy subset of and, the set is called the level subset of [1,6,13,25].

Definition 5.1 A fuzzy subset of an (m, n)-semihyperring is called a fuzzy (m, n)-sub-semihyperring of if following hold true:

1)

for all

2)

for all.

Definition 5.2 A fuzzy subset of an (m, n)-semihyperring is called a fuzzy hyperideal of if the following hold true:

1)

for all

2), for all 3), for all ,

4), for all .

Theorem 5.3 A fuzzy subset of an (m, n)-semihyperring is a fuzzy hyperideal if and only if every non-empty level subset is a hyperideal of.

Proof. Suppose subset is a fuzzy hyperideal of (m, n)-semihyperring and is a level subset of.

If for some then from the definition of level set, we can deduce the following:

Thus, we say that:

Thus:

(21)

So, we get the following:

, for all.

Therefore,.

Again, suppose that and, where. Then, we find that.

So, we obtain the following:

(22)

Thus, we find that is a hyperideal of.

On the other hand, suppose that every non-empty level subset is a hyperideal of.

Let, for all .

Then, we obtain the following:

Thus,

We can also obtain that:

Thus,

(23)

Again, suppose that. Then.

So, we obtain:

Thus,.

Similarly, we obtain, for all.

Thus, we can check all the conditions of the definition of fuzzy hyperideal.

This proof is a generalization of [1].

Theorem 5.3 is a generalization of [1,11,26].

Jun, Ozturk and Song [27] have proposed a similar theorem on hemiring.

Theorem 5.4 Let be a non-empty subset of an (m, n)-semihyperring. Let be a fuzzy set defined as follows:

where. Then is a fuzzy left hyper ideal of if and only if is a left hyper ideal of.

Following Corollary 5.5 is generalization of [1].

Corollary 5.5 Let be a fuzzy set and its upper bound be of an (m, n)-semihyperring. Then the following are equivalent:

1) is a fuzzy hyperideal of.

2) Every non-empty level subset of is a hyperideal of.

3) Every level subset is a hyperideal of where.

Definition 5.6 Let and be fuzzy (m, n)-semihyperrings and be a map from into. Then is called homomorphism of fuzzy (m, n)- semihyperrings if following hold true:

and

for all

Theorem 5.7 Let and be two fuzzy (m, n)-semihyperrings and and be associated (m, n)-semihyperring. If is a homomorphism of fuzzy (m, n)-semihyperrings, then is homomorphism of the associated (m, n)-semihyperrings also.

Definition 5.6 and Theorem 5.7 are similar to the one proposed by Leoreanu-Fotea [16] on fuzzy (m, n)-ary hyperrings and (m, n)-ary hyperrings and Ameri and Nozari [8] proposed a similar Definition and Theorem on hyperalgebras.

6. Conclusion

We proposed the definition, examples and properties of (m, n)-semihyperring. (m, n)-semihyperring has vast application in many of the computer science areas. It has application in cryptography, optimization theory, fuzzy computation, Baysian networks and Automata theory, listed a few. In this paper we proposed Fuzzy (m, n)- semihyperring which can be applied in different areas of computer science like image processing, artificial intelligence, etc. We found some of the interesting results: the natural map from an (m, n)-semihyperring to the quotient of the (m, n)-semihyperring is an onto homomorphism. It is also found that if and are two congruence relations on (m, n)-semihyperring such that, then is a congruence on and We found many interesting results in fuzzy (m, n)-semihyperring as well, like, a fuzzy subset of an (m, n)-semihyperring is a fuzzy hyperideal if and only if every non-empty level subset is a hyperideal of. We can use (m, n)-semihyperring in cryptography in our future work.

7. Acknowledgements

The first author is indebted to Prof. Shrisha Rao of IIIT Bangalore for encouraging him to do research in this area. A few basic definitions were presented when the first author was a master’s student under the supervision of Prof. Shrisha Rao at IIIT Bangalore.

REFERENCES