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 Ai and Bi 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.