Lateral Hyperbases and Covered Lateral Hyperideals of Ordered Ternary Semihypergroups

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1783 | P a g e

Lateral Hyperbases and Covered Lateral Hyperideals of

Ordered Ternary Semihypergroups

M.Y. Abbasi

1

, Sabahat Ali Khan

2

, Aakif Fairooze Talee

3

1,2,3

Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, (India)

ABSTRACT

In this paper, we define lateral hyperbases and covered lateral hyperideals of ordered ternary semihypergroups.

Here we study their related properties and explore the relationship between lateral hyperbases and covered

lateral hyperideals of ordered ternary semihypergroups.

Keywords:Ordered ternary semihypergroup, Lateral hyperbase, Covered lateral hyperideal.

AMS Mathematics Subject Classification (2010):

20M12, 20N99.

I. INTRODUCTION AND PRELIMINARIES

The idea of investigation of

n

-ary algebras i.e. the sets with one

n

-ary operation was given by Kasner’s [15]. In particular,

n

-ary semigroups are known as ternary semigroups for

n

=3 with one associative operation [17]. Ideal theory in ternary semigroup was studied by Sioson [21]. He also defined regular ternary semigroups. Marty [18] proposed the notion of algebraic hyperstructures as an extension of the branch of classical algebraic structures. In algebraic structures, the composition of two elements is an element, while in algebraic hyperstructure the composition of two elements is a non-empty set. Ternary semihypergroups are known as

n

-ary semihypergroup for

n

=

3

with one ternary associative hyperoperation. Davvaz and Leoreanu [10] studied binary relations on ternary semihypergroups and studied some basic properties of compatible relations on them. Hila and Naka [20] defined the notion of regularity in ternary semihypergroups and characterize them by using various hyperideals of ternary semihypergroups. In [19] they gave some properties of left (right) and lateral hyperideals in ternary semihypergroups. Hila et al.[15] introduced some classes of hyperideals in ternary semihypergroups.

The concept of ordering hypergroups investigated by Chavlina [4] as a special class of hypergroups. Heidari and Davvaz [13] studied a semihypergroup

(

S

,

o

)

besides a binary relation



, where



is a partial order relation

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1784 | P a g e

ordered ternary semihypergroup via hyperideals.

Tamura [23] introduced the notion of (right)left base of semigroup. Later, Fabrici [18] described the structure of semigroup containing one-sided bases. Summaprad and Changphas [22] defined the relationship between right bases and maximal right ideals.

Fabrici [14] showed some other properties and the mutual relation between covered ideals and bases of semigroups. Changphas and Summaprab [23] studied ordered semigroup containing covered ideals. They generalised the results on semigroup studied by Fabrici [12]. Basar et. al. [1] studied some properties of covered



-ideals in po- -semigroups. Thongkam and Changphas [24] introduced the notion of left bases and right

bases of a ternary semigroup

First, we recall some basic terms and definitions:

Definition 1.1 [17] A non-empty set

T

is called a ternary semigroup if there exists a ternary operation

T





, written as

(

x

₁

,

x

₂

,

x

₃

)



[

x

₁

x

₂

x

₃

]

, satisfying the following identity for any 5

4 3 2

1

,

x

,

x

,

x

,

x



T

,

]]

[

=

]

[

=

]

[[

x

1

x

2

x

3

x

4

x

5

x

1

x

2

x

3

x

4

x

5

x

1

x

2

x

3

x

4

x

5 .

For non-empty subsets

A

,

B

and

C

of a ternary semigroup

T

,

B

b

A

a

abc

ABC

]

:=

{[

]

:



,



[

and

c



C

}

.

If

A

=

{

a

}

, then we write

[{

a

}

BC

]

as

[

aBC

]

and similarly if

B

=

{

b

}

or

C

=

{

c

}

, we write

[

AbC

]

and

]

[

ABc

, respectively. Throughout the paper, we denote

[

x

1

x

2

x

3

]

by

x

1

x

2

x

3 and

[

ABC

]

as

ABC

.

For any positive integers

m

and

n

with

m



n

be any elements

x

₁

,

x

₂

,

x

₃

...

x

₂_n and

x

₂_n_₁ of a ternary semigroup [21], written as

]

]...

]

..[[

,

[

=

]

...

[

x

1

x

2

x

3

x

2n1

x

1

x

2

x

3

x

m

x

m1

x

m2

x

m3

x

m4

x

2n1 .

Definition 1.2 A ternary hypergroupoid is called the pair

(

H

,

f

)

where

f

is a ternary hyperoperation on the

set

H

.

If

A

,

B

,

C

are non-empty subsets of

H

, then we define

)

,

(

A

B

C

f

=



C c B b A a , ,

)

,

(

a

b

c

f

.

Definition 1.3 [10] A ternary hypergroupoid

(

H

,

f

)

is called a ternary semihypergroup if for all



3 2 1

,

a

,...,

a

H

, we have

))

,

(

,

(

=

)

),

,

(

,

(

=

)

,

),

,

(

f

a

1

a

2

a

3

a

4

a

5

f

a

1

f

a

2

a

3

a

4

a

5

f

a

1

a

2

f

a

3

a

4

a

5

f

.

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1785 | P a g e

It is clear that due to the associative law in ternary semihypergroup

H

, for any elements

x

1

,

x

2

,...,

x

2n1



H

and positive integers

m

,

n

with

m



n

, one may write,

).

),...,

,

),

,

(

,...,

(

=

)

,...,

,

,...,

(

=

)

,...,

,

(

1 2 4 3 2 1 1

1 2 2 1 1

1 2 2 1

 

  

 

n m

m m m m

n m

m m n

x

f

x

f

x

f

x

f

Definition 1.4 [10] Let

(

H

,

f

)

be a ternary semihypergroup. A binary relation



is called:

• compatible on the left if

a



b

and

x



f

(

x

₁

,

x

₂

,

a

)

imply that there exists

y



f

(

x

₁

,

x

₂

,

b

)

such that

x



y

;

• compatible on the right if

a



b

and

x



f

(

a

,

x

₁

,

x

₂

)

y



f

(

b

,

x

₁

,

x

₂

)

such that

x



y

;

• compatible on the lateral if

a



b

and

x



f

(

x

₁

,

a

,

x

₂

)

y



f

(

x

₁

,

b

,

x

₂

)

such that

x



y

;

• compatible on the two-sided if

a

₁



b

₁,

a

₂



b

₂, and

x



f

(

a

₁

,

z

,

a

₂

)

y



f

(

b

₁

,

z

,

b

₂

)

such that

x



y

;

• compatible if

a

₁



b

₁,

a

₂



b

₂,

a

₃



b

3 and

x



f

(

a

1

,

a

2

,

a

3

)

y



f

(

b

₁

,

b

₂

,

b

₃

)

such that

x



y

.

Definition 1.5 [2] A ternary semihypergroup

(

H

,

f

)

is called a partially ordered ternary semihypergroup if

there exits a partially ordered relation

'



on

H

such that

'



are compatible on left, compatible on right, compatible on lateral and compatible.

Example 1.1 Let

H

=

{













j

i

h

g

f

e

d

c

b

a

0

0 }

,

:

a

b

c

d

e

f

g

h

i

j



N

₀ , where

N

₀ i.e. the set of all

non-negative integers is an ordered ternary semigroup under the ordinary multiplication of numbers with

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1786 | P a g e

Now, let

A

₁ =

{













b

a

0

0 }

,

:

a

b



N

0 ,

A

2 =

{













0

0 d

c

b

a

}

,

:

a

b

c

d



N

0 .

Then

H

is a ternary semihypergroup under ternary hyperoperation

'

f



defined as follows:

)

,

(

X

Y

Z

f

=

X



A

₁



Y



A

₂



Z

for all

X

,

Y

,

Z



H

, where

'



is the usual multiplication of matrices over

N

₀.

Now we define partial order relation



_H on

H

by, for any

A

,

B



H

A



_H

B

if and only if

a

_ij



_N

b

_ij, for all

i

and

j

.

Then it is easy to verify that

H

is an ordered ternary semihypergroup with partial order relation



_H.

Throughout the paper, we denote

(

H

,

f

,



)

as an ordered ternary semihypergroup.

Lemma 1.1 [2] For subsets

A

,

B

and

C

of

(

H

,

f

,



)

, the following statements hold: •

A



(

A

]

for every

A



H

.

• If

A



B

, then

(

A

]



(

B

]

for every

A

,

B



H

.

•

((

A

]]

=

(

A

]

for every

A



H

.

•

f

((

A

],

(

B

],

(

C

])



(

f

(

A

,

B

,

C

)]

.

•

(

f

((

A

],

(

B

],

(

C

])]



(

f

(

A

,

B

,

C

)]

for all

A

,

B

,

C



H

.

(

H

,

f

,



)

be an ordered ternary semihypergroup and





T



H

. Then

T

is called an ordered ternary sub-semihypergroup of

H

if and only if

f

(

T

,

T

,

T

)



T

and (T]



T

.

Definition 1.7 [2] An element

a

of an ordered ternary semihypergroup

(

H

,

f

,



)

is called regular if there exists an element

x

in

H

such that

a



(

f

(

a

,

x

,

a

)]

.

H

is called regular ordered ternary semihypergroup if every element of

H

is regular.

Definition 1.8 [2] A non-empty subset

I

(

H

,

f

,



)

is called a right hyperideal(resp., lateral hyperideal, left hyperideal) of

H

if

•

f

(

I

,

H

,

H

)



I

(

f

(

H

,

I

,

H

)



I

,

f

(

H

,

H

,

I

)



I

);

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1787 | P a g e

Definition 1.9 [2] A non-empty subset

I

of

(

H

,

f

,



)

is called a two sided hyperideal of

H

if it is a left, right hyperideal of

H

and

I

is called hyperideal of

H

if it is a left, right and lateral hyperideal of

H

.

Lemma 1.2 [2] Let

(

H

,

f

,



)

be an ordered ternary semihypergroup. For any





A



H

, •

(

f

(

A

,

H

,

H

)•



A

]

is the smallest right hyperideal of

H

containing

A

;

•

(

f

(

H

,

H

,

A

,

H

,

H

)•



f

(

H

,

A

,

H

)



A

]

is the smallest lateral hyperideal of

H

containing

A

; •

(

f

(

H

,

H

,

A

)•



A

]

is the smallest left hyperideal of

H

containing

A

;

•

(

f

(

A

,

H

,

H

)



f

(

H

,

H

,

A

,

H

,

H

)



f

(

H

,

A

,

H

)



f

(

H

,

H

,

A

)



A

]

is the smallest hyperideal of

H

containing

A

.

Lemma 1.3 [2] Let

(

H

,

f

,



)

be an ordered ternary semihypergroup. For any





A



H

, •

(

f

(

A

,

H

,

H

)]

is a right hyperideal of

H

;

•

(

f

(

H

,

H

,

A

,

H

,

H

)



f

(

H

,

A

,

H

)]

is a lateral hyperideal of

H

; •

(

f

(

H

,

H

,

A

)]

is a left hyperideal of

H

;

•

(

f

(

A

,

H

,

H

)



f

(

H

,

H

,

A

,

H

,

H

)



f

(

H

,

A

,

H

)



f

(

H

,

H

,

A

)]

is a hyperideal of

H

.

(

H

,

f

,



)

be an ordered ternary semihypergroup. A lateral hyperideal

M

of

H

is called a minimal lateral hyperideal of

H

if there is no lateral hyperideal

M

'of

H

such that

M

'



M

.

Definition 1.11 [2] Let

(

H

,

f

,



)

be a ternary semihypergroup. A right hyperideal

R

of

H

is called a maximal right hyperideal of

H

if for every right hyperideal

A

of

H

such that

R



A

, we have

A

=

H

.

Let

H

be an ordered ternary semihypergroup and

a

,

b

be any non-zero elements of

H

. Then the equivalence relation

M

is defined by:

a

M

b

if

a

and

b

generate the same principal lateral hyperideal of

T

.

M

-class containing

a

is denoted by

M

_a. Now let

a

,

b

be any non-zero elements f

H

, then we define a

quasi-ordering on the set of all equivalence classes as:

M

_a

M

_b if donly if

M

(

a

)



M

(

b

)

, where

M

(

a

)

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1788 | P a g e

II. LATERAL HYPERBASES OF ORDERED TERNARY SEMIHYPERGROUPS

Definition 2.1 Let

(

H

,

f

,



)

be an ordered ternary semihypergroup. A non-empty subset

A

of

H

is called a lateral hyperbase of

H

if,

(1)

(

A



f

(

H

,

A

,

H

)



f

(

H

,

H

,

A

,

H

,

H

)]

=

H

, and

(2)

(

B



f

(

H

,

B

,

H

)



f

(

H

,

H

,

B

,

H

,

H

)]



H

, for any proper subset

B

of

A

.

Example 2.1 Let

H

={

















0

0 d

c

b

a

}

,

:

a

b

c

d



N

₀ , where

N

₀ i.e. the set of all non-negative integers is

an ordered ternary semigroup under the ordinary multiplication of numbers with partial ordered relation



_N

is "less than or equal to".

Then

H

is a ternary semihypergroup under ternary hyperoperation

'

f



defined as follows:

)

,

(

X

Y

Z

f

=

X



Y



Z

for all

X

,

Y

,

Z



H

,

where

'



N

₀. Also,

H

is an ordered ternary semihypergroup under

f

over

N

₀ with partial order relation



_H, as defined in Example 1.1.

Now consider a set

M

=

{

















0

1

0

1

0 }

. Then, we can easily verify that

M

is lateral hyperbase of

H

as

)]

,

(

)

,

(

M



f

H

M

H



f

H

M

H

=

H

.

Lemma 2.1 Let

(

H

,

f

,



)

be an ordered ternary semihypergroup.. If two elements

a

,

b

belongs to lateral hyperbase

A

of

H

such that

a



(

f

(

H

,

b

,

H

)



f

(

H

,

H

,

b

,

H

,

H

)]

. Then

a

=

b

.

Proof. Let

A

be a lateral hyperbase of an ordered ternary semihypergroup

H

and

a

,

b



A

such that

a



(

f

(

H

,

b

,

H

)



f

(

H

,

H

,

b

,

H

,

H

)]

. Suppose

a



b

. Take a proper subset

B

of

A

as

B

=

A

\

{

a

}

. Then

b

belongs to

B

. Now

a



(

f

(

H

,

b

,

H

)



f

(

H

,

H

,

b

,

H

,

H

)]

, it implies

).

(

)

(

)]

,

(

)

,

(

)

(

B

b

f

b

f

a

M

H

M







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1789 | P a g e

H

. Therefore,

a

=

b

.

Lemma 2.2Let

A

be a subset of an ordered ternary semihypergroup

(

H

,

f

,



)

. Then

A

is a lateral hyperbase of

H

if and only if it satisfies the following conditions:

(1) For any

t



H

there exists

a

in

A

such that

M

_t

M

_a;

(2) If

a

₁,

a

₂



A

such that

a

₁



a

2, then neither

M

a₁

M

a₂ nor

M

a₂

M

a₁.

Proof. Suppose

A

is a lateral hyperbase of an ordered ternary semihypergroup

(

H

,

f

,



)

. Let

t



H

. As

A

is a lateral hyperbaseof

H

, we have

(

A



f

(

H

,

A

,

H

)



f

(

H

,

H

,

A

,

H

,

H

)]

=

H

. Then

t



(

A



f

(

H

,

A

,

H

)



f

(

H

,

H

,

A

,

H

,

H

)]

, so

t



(

A

]

or

t



(

f

(

H

,

A

,

H

)]

or

t



(

f

(

H

,

H

,

A

,

H

,

H

)]

. If

t



(

A

]

, then

t



a

, for some

a



A

, it follows

M

_t

M

_a. If

t



(

f

(

H

,

A

,

H

)]

, then

t



f

(

t

₁

,

a

'

,

t

₂

)

for some

t

₁

,

t

₂



H

and

a

'



A

, we have

M

_t _'

a

M

. If

t



(

f

(

H

,

H

,

A

,

H

,

H

)]

, then

t



f

(

t

₃

,

t

₄

,

a

'

,

t

₅

,

t

₆

)

for some

t

₃

,

t

₄

,

t

₅

,

t

₆



H

and

a

'



A

, we obtain

M

_t _'_ a

M

. Hence, the result (i)

is proved.

Now, suppose

a

₁



a

₂ i.e.

1 a

M

2 a

M

,

M

(

a

₁

)



M

(

a

₂

)

. If

a

₁



a

₂,

then

a

₁



(

f

(

H

,

a

2

,

H

)



f

(

H

,

H

,

a

2

,

H

,

H

)]

. So, by Lemma 2.1,

a

1

=

a

2. Thus for

a

1



a

2,

M

a₁

M

a₂. Analogously, we can prove

2 a

M

1 a

M

.

Conversely, suppose that conditions (i) and (ii) holds. From (i),

M

(

A

)

=

H

. Now, if possible

M

(

B

)

=

H

,

where

B

is a proper subset of

A

. Let

a

₁



A

\

B

, then there exists

a

₂



B

such that 1

a



(

a

₂



f

(

H

,

a

₂

,

H

)



f

(

H

,

H

,

a

₂

,

H

,

H

)]

. Hence

M

(

a

₁

)



M

(

a

₂

)

, i.e. 1 a

M

2 a

M

, which is a

contradiction to the assumption (2). Therefore,

M

(

B

)



H

. Hence,

A

is a lateral hyperbase of

H

.

Theorem 2.1 Let an ordered ternary semihypergroup

(

H

,

f

,



)

contains a lateral hyperideal. Then

M

is a maximal lateral hyperideal of

H

if and only if

H

\

M

is a maximal

M

-class.

Proof. Suppose

M

is a maximal lateral hyperideal of an ordered ternary semihypergroup

(

H

,

f

,



)

. Let

t

₁, 2

t



H

\

M

. As

M



M



M

(

t

₁

)



H

, it implies

M



M

(

t

₁

)

=

H

. Thus,

t

₂



M

(

t

₁

)

. Similarly 1

(8)

1790 | P a g e

maximal

M

-class.

Conversely, suppose that

H

\

M

is a maximal

M

-class such that

H

\

M

=

M

_t for some

t



H

. We have to show that

M

is maximal lateral hyperideal of

H

. Firstly, we will prove that

M

H

. i.e.

)

,

(

H

M

H

f



f

(

H

,

H

,

M

,

H

,

H

)



M

and

(

M

]

=

M

. On contrary suppose that

M

is not a lateral hyperideal of

H

. Then there exists

m



f

(

H

,

M

,

H

)



f

(

H

,

H

,

M

,

H

,

H

)

such that

m



M

. It implies

m



H

\

M

=

M

_t, then we have

M

(

m

)

=

M

(

t

)

. As

m



f

(

H

,

M

,

H

)



f

(

H

,

H

,

M

,

H

,

H

)

, there exists 1

t

,

t

₂,...,

t

₆



H

and

m

₁,

m

₂



M

such that

m



f

(

t

₁

,

m

₁

,

t

₂

)

or

m



f

(

t

3

,

t

4

,

m

2

,

t

5

,

t

6

)

. Hence, we have

M

_t



1 m

M

or

M

_t



2 m

M

. Therefore,

M

_t



M

m, which is a contradiction to the assumption that

H

\

M

is a maximal

M

-class. It implies

f

(

H

,

M

,

H

)



f

(

H

,

H

,

M

,

H

,

H

)



M

. Now, let

m



M

and

s



H

such that

s



m

. Suppose

s



H

\

M

=

M

_t. Then

s

<

m

, hence

M

_t



M

_m. This is a contradiction. Thus, we have

s



M

. Therefore,

M

H

. Let

M

' be a proper lateral hyperideal of

H

such that

M

is properly contained in

M

'. Then there exists

a



H

\

M

'. Thus,

M

_t

=

M

a. Also there exists

'

m



M

'

\

M

such that _' m

M

=

M

_t. It follows that

a



M

_a=

M

_t = _' m

M



M

', which is also a contradiction.

Therefore,

M

is a maximal lateral hyperideal of

H

.

III. COVERED LATERAL HYPERIDEALS OF ORDERED TERNARY

SEMIHYPERGROUPS

Definition 3.1 A proper lateral hyperideal

L

(

H

,

f

,



)

is said to be covered lateral hyperideal (CLt. ideal) if

L



(

f

(

H

,

(

H



L

),

H

)



f

(

H

,

H

,

(

H



L

),

H

,

H

)]

.

Remark 3.1 By definition of CLt.-hyperideal, ternary semihypergroup itself is not a CLt.-hyperideal.

Example 3.1 Let

H

=

{

















f

e

d

c

b

a

0

0 }

,

:

a

b

c

d

e

f



N

₀ excluding null matrix, where

N

₀ is an

ordered ternary semigroup under the ordinary multiplication of numbers with partial ordered relation 0 N



is

"less than or equal to".

(9)

1791 | P a g e

)

,

(

X

Y

Z

f

=

X



Y



Z

for all

X

,

Y

,

Z



H

,

where

'



N

₀. Then

H

is an ordered ternary semihypergroup under

f



'

over

N

₀ with partial order relation



_H, as defined in the Example 1.1.

Consider

L

=

{

















0

0 b

:

b



N

}

. Then, it easy to verify that

L

is lateral hyperideal of

H

and

H



L

=

{

















e

d

c

b

a

0

0 }

,

:

a

b

c

d

e



N

0 .

Now,

L

=

{

















0

0 b

:

b



N

}



{

















f

e

d

c

b

a

0

0 }

,

:

a

b

c

d

e

f



N

0 =

)

),

(

,

(

f

H



L

H



f

(

H

,

H

,

(

H



L

),

H

,

H

)]

. Therefore,

L

is a CLt.-hyperideal of

H

.

Lemma 3.1 Let

(

H

,

f

,



)

be an ordered ternary semihypergroup and let

H

contains two different lateral hyperideals

L

₁ and

L

₂ such that

L

₁



L

₂=

H

, then none of the lateral hyperideals

L

₁,

L

₂ is a CLt.-hyperideal.

Proof. Suppose

(

H

,

f

,



)

is an ordered ternary semihypergroup and

L

₁,

L

₂ are two different lateral

hyperideals of

H

such that

L

₁



L

₂ =

H

. It implies

H

-

L

₁



L

₂ and

H

-

L

₂



L

₁. Suppose that

L

₂ is a covered lateral hyperideal of

H

, then

L

₂



(

f

(

H

,

(

H



L

₂

),

H

)



f

(

H

,

H

,

(

H



L

₂

),

H

,

H

)]

which implies

2

L



(

f

(

H

,

(

H



L

₂

),

H

)



f

(

H

,

H

,

(

H



L

₂

),

H

,

H

)]



(

f

(

H

,

L

₁

,

H

)



f

(

H

,

H

,

L

₁

,

H

,

H

)]



)

,

(

f

H

L

₁

H



f

(

H

,

L

₁

,

H

)]



L

₁. Similarly

L

₁



L

₂. Therefore,

L

₁ =

L

₂. But

L

₁ and

L

₂ are different. Thus, our assumption was wrong. Hence, neither

L

₁ nor

L

₂ is a CLt.-hyperideal.

Corollary 3.1 Let

(

H

,

f

,



)

be an ordered ternary semihypergroup. If an ordered ternary semihypergroup

H

contains more than one maximal lateral hyperideal, then none of the maximal lateral hyperideal is

(10)

1792 | P a g e

Proof. Suppose that the ordered ternary semihypergroup

H

contains two maximal lateral hyperideals

M

₁ and

2

M

. We know that union of lateral hyperideals is a lateral hyperideal. Then

M

₁



M

₂ is a lateral hyperideal of

H

and

M

₁



M

₁



M

₂. As

M

₁ is a maximal lateral hyperideal of

H

. It implies

M

₁



M

₂ =

H

. Hence, by Lemma 3.1, neither

M

₁ nor

M

₂ is a CLt.-hyperideal of

H

.

Lemma 3.2 Let

(

H

,

f

,



)

be an ordered ternary semihypergroup. If

L

is a lateral hyperideal of

H

such that

L



(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]

and

L



(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]

for some

t



H

. Then

L

will be a CLt. hyperideal of

H

.

Proof. Suppose that

(

H

,

f

,



)

is an ordered ternary semihypergroup. Let

L

be a lateral hyperideal of

H

such that

L



(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]

and

L



(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]

for some

t



H

. Here

t



L

, otherwise

(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]



(

f

(

H

,

L

,

H

)



f

(

H

,

H

,

L

,

H

,

H

)]



L

and we assume that

L



(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]

. Hence

L



(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]



(

f

(

H

,

(

H



L

),

H

)



f

(

H

,

H

,

(

H



L

),

H

,

H

)]

. This shows that

L

is a CLt. hyperideal of

H

.

Corollary 3.2 An ordered ternary semihypergroup

H

in which

t

does not belongs to

)

,

(

f

H

t

H



f

(

H

,

H

,

t

,

H

,

H

)]

contains some CLt.-hyperideal.

Proof. Let

L

=

(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]

. Then

L

H

. If

t



L

, we have

L

=

(

f

(

H

,

t

,

H

)



f

(

H

,

H

,

t

,

H

,

H

)]



(

f

(

H

,

(

H



L

),

H

)



f

(

H

,

H

,

(

H



L

),

H

,

H

)]

. This implies

L

is a CLt.-hyperideal of

H

.

Lemma 3.3 Let

(

H

,

f

,



)

be an ordered ternary semihypergroup and

L

₁ and

L

₂ be two covered lateral hyperideal of

H

. Then

L

₁



L

₂ is a CLt.-hyperideal of

H

.

Proof. Assume

(

H

,

f

,



)

be an ordered ternary semihypergroup and

L

₁ and

L

₂ are two covered lateral hyperideal of

H

. Now to prove

L

₁



L

H

, we have to show that

1

(11)

1793 | P a g e

2

1

,

m



H



L

₁ such that

m



(

f

(

H

,

m

₁

,

H

)



f

(

H

,

H

,

m

₂

,

H

,

H

)]

. Now we have following four cases:

Case1: If

m

₁

,

m

₂



H



L

₁



L

₂. Then

m



(

f

(

H

,

(

H



L

₁



L

₂

),

H

)



f

(

H

,

H

,

(

H



L

₁



L

₂

),

H

,

H

)]



(

f

(

H

,

[

H



(

L

₁



L

₂

)],

H

)



)]

,

)],

(

[

,

(

H



L

₁



L

₂

H

f

.

Case2: If

m

₁

,

m

₂



(

H



L

₁

)



L

₂. It implies 2

1

,

m



L

₂



(

f

(

H

,

(

H



L

₂

),

H

)



f

(

H

,

H

,

(

H



L

₂

),

H

,

H

)]

. Then, there exists 6

5 4 3

,

m

,

m

,

m



H



L

₂ s.t.

m

₁



(

f

(

H

,

m

₃

,

H

)



f

(

H

,

H

,

m

₄

,

H

,

H

)]

and 2

m



(

f

(

H

,

m

₅

,

H

)



f

(

H

,

H

,

m

₆

,

H

,

H

)]

. Here

m

₃

,

m

₄



L

₁, otherwise 1

m



(

f

(

H

,

m

₃

,

H

)



f

(

H

,

H

,

m

₄

,

H

,

H

)]



(

f

(

H

,

L

₁

,

H

)



f

(

H

,

H

,

L

₁

,

H

,

H

)]



L

₁. Hence

m

₁



L

₁, which is contradiction as

m

₁



H



L

₁. Thus we have

m

₃

,

m

₄



H



L

₁. Therefore

4 3

,

m



H



L

₁



H



L

₂ =

H



(

L

₁



L

₂

)

. Similarly

m

₅

,

m

₆



H



(

L

₁



L

₂

)

. Now

)].

,

)],

(

[

,

(

)

)],

(

[

,

(

)]

,

(

)

,

(

)

,

(

)

,

(

)]

,

),

,

(

)

,

(

,

(

)

),

,

(

)

,

(

,

(

=

)]]

,

),

,

(

)

,

(

,

(

)

),

,

(

)

,

(

,

(

((

])]

(

],

(

)],

,

(

)

,

(

],

(

],

((

])

(

)],

,

(

)

,

(

],

((

(

)]

,

)],

,

(

)

,

(

,

(

)

)],

,

(

)

,

(

,

(

)]

,

(

)

,

(

2 1 2 1 6 5 4 3 6 5 4 3 6 5 4 3 6 5 4 3 6 5 4 3 2 1

H

L

H

L

H































f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

f

m

Case 3: If

m

₁



H



L

₁



L

₂ and

m

₂



(

H



L

₁

)



L

₂. As

m

₁



H



L

₁



L

₂, it implies 1

m



H



(

L

₁



L

₂

)

. From case 2,

m

₂



H



(

L

₁



L

₂

)

. Therefore

m



(

f

(

H

,

m

₁

,

H

)



f

(

H

,

H

,

m

₂

,

H

,

H

)]



)]

,

)],

(

[

,

(

)

)],

(

[

,

(

f

H



L

₁



L

₂

H



f

H



L

₁



L

₂

H

.

Case 4: If

m

₂



H



L

₁



L

₂ and

m

₁



(

H



L

₁

)



L

₂. Then this is similar to case 3.

Therefore, in all these cases

m



(

f

H

,

[

H



(

L

₁



L

₂

)],

H

)



f

(

H

,

H

,

[

H



(

L

₁



L

₂

)],

H

,

H

)]

. Similarly, we can prove this for

m



L

₂.

(12)

1794 | P a g e

CLt.-hyperideal of

H

.

Lemma 3.4 Let

(

H

,

f

,



)

L

₁ is a covered lateral hyperideal of

H

and

L

₂ is any lateral hyperideal of

H

. Then,

L

₁



L

H

, provided

L

₁



L

₂

 

.

Proof. Assume that

L

₁ is a covered lateral hyperideal and

L

₂ is a lateral hyperideal of

H

such that

1

L



L

₂

 

. Then

L

₁



(

f

(

H

,

(

H



L

₁

),

H

)



f

(

H

,

H

,

(

H



L

₁

),

H

,

H

)]

. It implies

)].

,

)],

(

[

,

(

)

)],

(

[

,

(

)]

,

),

(

,

(

)

),

(

,

(

2 1 2

1

1 1

2 1

H

L

H

L

H

L

H

L

H

L























f

Therefore,

L

₁



L

H

.

Lemma 3.5 Let

(

H

,

f

,



)

L

₁ and

L

₂ are two covered lateral hyperideal of

H

. Then,

L

₁



L

H

, provided

L

₁



L

₂

 

.

Proof. Proof is similar to the above lemma.

Definition 3.2 A proper lateral hyperideal

L

is said to be the greatest lateral hyperideal of on ordered ternary

semihypergroup

H

if

L

contains any proper lateral hyperideal of

H

. We shall denote it by

L

*.

Theorem 3.1 Let

(

H

,

f

,



)

H

have only one maximal lateral hyperideal

M

and

M

is a CLt.-hyperideal, then

M

=

M

*

Proof. Suppose that

M

is a maximal lateral of an ordered ternary semihypergroup

H

and

M

is also a

CLt.-hyperideal of

H

. Let

M

₁ be lateral hyperideal of

H

such that

M

₁



M

. As

M

₁



M

H

and

M



M

₁



M

. It follows that

M

₁



M

=

H

. Therefore, by Lemma 3.1,

H

cannot contains any CLt.-hyperideals, which is contradiction. Thus

M

₁



M

. Hence,

M

=

M

*.

Theorem 3.2 Let

(

H

,

f

,



)

H

is not a simple such that there is no any two proper lateral hyperideals in which there intersection is empty. Then

H

contains at least one

CLt.-hyperideal.