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
![](https://www.scirp.org/html/12-1100194\15b1befc-bdf6-4119-a8cc-2dfb39907cdc.jpg)
Let f be an m-ary hyperoperation on
and
subsets of
. We define
![](https://www.scirp.org/html/12-1100194\d1ed6d79-08b7-4b71-bed9-730d5984363c.jpg)
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) ![](https://www.scirp.org/html/12-1100194\c8dbf1c1-9319-4418-91ef-390064f29001.jpg)
2) ![](https://www.scirp.org/html/12-1100194\e9c45efe-4add-4509-be95-6ebedf293010.jpg)
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:
![](https://www.scirp.org/html/12-1100194\7095470a-f7df-4c4e-9bc6-525d725a51c5.jpg)
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:
![](https://www.scirp.org/html/12-1100194\40b06af4-375d-4adb-9fad-0f579c38346f.jpg)
for all
and
[14].
Definition 2.4 Element e is called identity element of hypergroup
if
![](https://www.scirp.org/html/12-1100194\21badcae-e49f-46b1-a06f-4c297f5b4f0a.jpg)
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.,
![](https://www.scirp.org/html/12-1100194\75b888ad-f9b1-4cb2-b05b-bff738641f4f.jpg)
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.,
![](https://www.scirp.org/html/12-1100194\e97a91fd-160d-4038-bc5d-2e29062fe7cb.jpg)
Remark 3.2 An (m, n)-semihyperring is called weak distributive if it satisfies Definition 3.1 1), 2) and the following:
![](https://www.scirp.org/html/12-1100194\7a564bee-313e-42bc-97f2-3bcfd04d168f.jpg)
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:
![](https://www.scirp.org/html/12-1100194\f86f46b8-8c1f-4cc1-8269-830e061f0dee.jpg)
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
![](https://www.scirp.org/html/12-1100194\be54f661-0a16-4492-bae2-30c3ccb7b31b.jpg)
for all
and
and
![](https://www.scirp.org/html/12-1100194\91b7da93-8132-4e43-a56b-58f1e2d13278.jpg)
for all
and
.
Definition 3.5 An element 0 of an (m, n)-semihyperring
is called a zero of
if
![](https://www.scirp.org/html/12-1100194\1ad6cd10-2493-4db6-a613-91ea9f78c6fd.jpg)
for all
.
![](https://www.scirp.org/html/12-1100194\759cc9f7-8df1-4e61-bc33-d38d2bc3d44d.jpg)
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:
![](https://www.scirp.org/html/12-1100194\30f15eea-3fbf-448c-beac-c3101ee79aea.jpg)
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
,
.
![](https://www.scirp.org/html/12-1100194\b811c12f-8188-4501-a238-75e889d2b152.jpg)
and
![](https://www.scirp.org/html/12-1100194\a4f65d79-b277-4d5c-b729-593b25689f6e.jpg)
Remark 4.4 Let
and
be two (m, n)-semihyperrings. The mapping
for all
,
is called an inclusion homomorphism if following relations hold:
![](https://www.scirp.org/html/12-1100194\14e3cefc-8506-4792-8b40-03041578705c.jpg)
and
![](https://www.scirp.org/html/12-1100194\8bd513cb-11d3-41e7-a66a-8c460fd3ecb7.jpg)
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 ![](https://www.scirp.org/html/12-1100194\3b219d4b-99c9-4fa1-8a64-3cf0dccbe021.jpg)
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
![](https://www.scirp.org/html/12-1100194\6dd951b8-ab3b-45c9-8344-06ad56bbe186.jpg)
for all ![](https://www.scirp.org/html/12-1100194\12ca492a-0b8b-453e-92c9-7b57ff5fc6aa.jpg)
2) if
then following holds:
![](https://www.scirp.org/html/12-1100194\cb5d129b-c3d8-4fca-8981-7e0449ded205.jpg)
for all ![](https://www.scirp.org/html/12-1100194\32da1aa2-6f51-4fac-9cef-1b098dd3f6b8.jpg)
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:
![](https://www.scirp.org/html/12-1100194\df98b522-a704-40c7-9e62-c2b57e12cbdb.jpg)
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 ![](https://www.scirp.org/html/12-1100194\a4ea5303-31b0-4de1-9eea-977fd3d7f76f.jpg)
![](https://www.scirp.org/html/12-1100194\09e9cfb9-6735-4b7e-8443-6d9a90c3d933.jpg)
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
![](https://www.scirp.org/html/12-1100194\e1b77610-7b40-41a0-94b1-de50110009d2.jpg)
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
![](https://www.scirp.org/html/12-1100194\ccc4f504-3395-4153-8bab-81027a124a9d.jpg)
is a congruence on
and ![](https://www.scirp.org/html/12-1100194\b3878fe0-3bfc-4b46-9888-ed9d94305c56.jpg)
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:
![](https://www.scirp.org/html/12-1100194\76f83c62-7f10-41c8-8d2d-552d40654fcf.jpg)
In a similar fashion we can deduce for
, for all
, where
:
![](https://www.scirp.org/html/12-1100194\cee4d042-6f60-4562-9cc0-6577314b9be6.jpg)
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) ![](https://www.scirp.org/html/12-1100194\107d6e91-87c7-413a-b6c5-3a734716c020.jpg)
for all ![](https://www.scirp.org/html/12-1100194\2442f5e4-f331-45b0-9ff3-93bb4120d5f7.jpg)
2) ![](https://www.scirp.org/html/12-1100194\50e956db-55df-459b-b87f-2bbe7167d6f3.jpg)
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) ![](https://www.scirp.org/html/12-1100194\3b3afe51-3a9a-427d-a0b4-5f3ee78a8252.jpg)
for all ![](https://www.scirp.org/html/12-1100194\0539b712-a5d3-4a18-b1de-7dcbfc6184b4.jpg)
2)
, for all
3)
, for all
,
![](https://www.scirp.org/html/12-1100194\4da04432-5a63-4bad-9321-d9491c38a890.jpg)
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:
![](https://www.scirp.org/html/12-1100194\e77ec59a-945a-486b-b6d5-61d1d07bb5ea.jpg)
Thus, we say that:
![](https://www.scirp.org/html/12-1100194\1847df6f-b68c-432c-92aa-f469f998e901.jpg)
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:
![](https://www.scirp.org/html/12-1100194\d4830b70-2f4c-4a68-abf9-74076a45177f.jpg)
Thus,
![](https://www.scirp.org/html/12-1100194\a42ad7b4-c76f-4e40-bc88-b640577eae08.jpg)
We can also obtain that:
![](https://www.scirp.org/html/12-1100194\8837b094-d485-4cb0-9cc8-033963026b32.jpg)
Thus,
(23)
Again, suppose that
. Then
.
So, we obtain:
![](https://www.scirp.org/html/12-1100194\e435d133-7302-4c56-9927-329a2b5b333e.jpg)
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:
![](https://www.scirp.org/html/12-1100194\a50abb7a-74e9-42b1-93c3-8a32aed563ad.jpg)
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:
![](https://www.scirp.org/html/12-1100194\5b779a48-dde6-4c26-b934-ed1db200cd34.jpg)
and
![](https://www.scirp.org/html/12-1100194\69be0cde-f35f-4662-ac84-33107fef3d5d.jpg)
for all ![](https://www.scirp.org/html/12-1100194\3a7aa99c-8aab-4eb3-ae2c-98e01a25b135.jpg)
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.