Songklanakarin J. Sci. Technol. 38 (2), 199-206, Mar. - Apr. 2016 http://www.sjst.psu.ac.th
Original Article
On (m, n)-ideals and (m, n)-regular ordered semigroups Limpapat Bussaban* and Thawhat Changphas Department of Mathematics, Faculty of Science, Khon Kaen University, Mueang, Khon Kaen, 40002 Thailand. Received, 1 July 2015; Accepted, 13 October 2015
Abstract Let m, n be non-negative integers. A subsemigroup A of an ordered semigroup (S, , is called an (m, n)-ideal of S if (i) AmSAn A, and (ii) if x A, y S such that y x, then y A. In this paper, necessary and sufficient conditions for every (m, n)-ideal (resp. (m, n)-quasi-ideal) of an (m, n)-ideal (resp. (m, n)-quasi-ideal) A of S is an (m, n)-ideal (resp. (m, n)-quasiideal) of S will be given. Moreover, (m, n)-regularity of S will be discussed. The results obtained extend the results on semigroups (without order) studied by Bogdanovic (1979). Keywords: semigroup, ordered semigroup, (m, n)-ideal, (m, n)-quasi-ideal, (m, n)-regular
1. Preliminaries Let m, n be non-negative integers. A subsemigroup A of a semigroup S is called an (m, n)-ideal of S if m
n
A SA A. Here, A0S = SA0 = S. This notion was first introduced and studied by Lajos (1961). Furthermore, the theory of (m, n)-ideals in other structures have also been studied by many authors (see also Akram et al., 2013; Amjad et al., 2014; Lajos, 1963; Yaqoob et al., 2012; Yaqoob et al., 2013; Yaqoob et al., 2014; Yousafzai et al., 2014). A semigroup S is said to be (m, n)-regular (Krgovic, 1975) if for any a in S, there exists x in S such that a a m xa n . Bogdanovic (1979) studied some properties of (m, n)-ideals and (m, n)-regularity of S. Indeed, the author characterized when every (m, n)-ideal of an (m, n)-ideal A of S is an (m, n)-ideal of S. Moreover, (m, n)regularity of S was discussed. In this paper, using the concepts of (m, n)-ideals and (m, n)-regularity of ordered semigroups introduced and studied by Sanboorisoot et al. (2012), we extend the results obtained by Bogdanovic (1979) mentioned above to ordered semigroups.
* Corresponding author. Email address:
[email protected],
[email protected]
A semigroup ( S , ) together with a partial order that is compatible with the semigroup operation, meaning that, for any a, b, c in S,
a b ac bc, ca cb, is said to be an ordered semigroup (Birkhoff, 1967; Fuchs, 1963). A non-empty subset A of an ordered semigroup ( S , , ) is said to be a subsemigroup of S if ab A for all a, b in A (Kehayopulu, 2006). If A and B are non-empty subsets of an ordered semigroup ( S , , ), the set product AB is defined to be the set of all elements ab S such that a A and b B , that is, AB {ab ∣a A, b B} . And, we write
( A] {x S ∣x a for some a A}. It is observed by Kehayopulu (2006) that the following conditions hold: (1) A ( A] ; (2) ( A]( B ] ( AB ] ; (3) If A B , then ( A] ( B] ; (4) ( A B] ( A] ( B ]; (5) ( A B ] ( A] ( B ] . A non-empty subset A of an ordered semigroup ( S , , ) is called a left (resp. right) ideal of S if it satisfies the following conditions: (i) SA A (resp. AS A ); (ii) ( A] A. And, A is called a two-sided ideal (or simply an ideal) of S if it is both a left and a right ideal of S (Kehayopulu, 2006). A subsemigroup B of S is called a bi-ideal of S if (i) BSB B ; (ii) ( B] B (Kehayopulu, 1992). A non-
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empty subset Q of S is called a quasi-ideal of S if (i) (QS ] ( SQ] Q ; (ii) (Q ] Q (Tsingelis, 1991; Kehayopulu, 1994). Note that if Q is a quasi-ideal of S, then it is a subsemigroup of S. In fact, if Q is a quasi-ideal of S, then QQ (QS ] ( SQ] Q . Finally, a subsemigroup A of S is called an (m, n)-ideal of S (m, n are non-negative integers) m n if (i) A SA A ; (ii) ( A] A (Sanborisoot et al., 2012). We first prove the following theorem.
mn
S (( Ai Am SAn ])n ( SAn ]. i 1
Then, mn
(( A A i
ing A of S. Then Ai is a subsemigroup containing A of iI S. For j I , we have
]
it follows that
mn i 1
i I
i I
( Ai ]
i I
Ai
(
i I
]
Ai ,
m
SAn ] (([ A]( m, n ) ) m S ([ A]( m , n ) ) n ] [ A]( m , n ) ,
mn
( A A i
m
SAn ] [ A]( m , n ) .
i 1
Ai is an (m, n)-ideal of S.
i
A Am SAn ] is a subsemigroup of S. We now
This completes the proof. For an element a of an ordered semigroup ( S , , ) , we write [{a}]( m , n ) (or simply [ a ]( m , n ) ) by: mn
consider:
[ a ]( m , n ) ( {a}i a m Sa n ] .
mn
(( A A i
m
i 1
SAn ]) m S
i 1 mn
m n
i 1
i 1
(( Ai Am SAn ])m 1 ( Ai Am SAn ]S m n
(( Ai Am SAn ])m 1 ( AS ] m n
(( A A SA i
i 1
mn
m
(ii) (Q ] Q .
m n
n
]) ( A m 2
i
A SA ]( AS ]
i 1
(( Ai Am SAn ]) m 2 ( A2 S ] i 1
( Am S ].
To extend the notion of (m, n)-quasi-ideals of semigroups defined by Lajos (1961), we introduce the concept of (m, n)-quasi-ideals of an ordered semigroup ( S , , ) as follows: let m, n be non-negative integers. A subsemigroup Q of S is called an (m, n)-quasi-ideal of S if it satisfies the following conditions: (i) (Q m S ] ( SQ n ] Q ;
i 1
Similarly,
i 1
it follows that
We will show that (1.1) holds. It is easy to see that
(
mn
n
Ai
i I
i
A Am SAn ] is an (m, n)-ideal containing A
mn i 1
[ A]( m , n ) ( Ai Am SAn ].
(A
Then ( iI Ai ) S ( iI Ai ) iI Ai . Since
(
(
Finally, by
Ai )m S ( iI Ai )n A mj SA nj A j . m
i 1
of S, and
Proof. Let { Ai | i I } be the set of all (m, n)-ideals contain-
iI
mn
( Ai Am SAn ].
(1.1)
i 1
(
i 1
( Am SAn ]
Hence
[ A]( m, n ) ( Ai Am SAn ]
SAn ]) m S (( Ai Am SAn ])n
i 1
Theorem 1.1. Let A be a non-empty subset of an ordered semigroup ( S , , ) . Then the intersection of all (m, n)-ideals containing A of S, denoted by [ A]( m , n ) , is an (m, n)-ideal containing A of S, and it is of the form m n
mn
m
m
n
0
0
Here, Q S SQ S. Note that every (m, n)-quasi-ideal of S is an (m, n)-ideal of S. It’s easy to see that if Q is a quasi-ideal of S, then Q is an (m, n)-quasi-ideal of S. The following example shows that an (m, n)-quasi-ideal of S needs not to be a quasi-ideal of S. Example 1.1. Let S {a, b, c, d } be an ordered semigroup with the multiplication and the order relation defined by:
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xy (
max{ m , n }
Ai ] (
max{ m , n}
i 1
Ai ] (( Am S ] (SAn ]).
i 1
If max{m, n} j k , then m, n j k , that is, ( A j k ] ( Am ( j k m ) ] ( Am S ] and ( A j k ] ( A( j k n ) n ] ( SAn ]. Hence max{ m , n }
xy ( A
j k
m
n
] ( A S ] ( SA ]
(
{(a, a ), (b, b), (c, c), (d , a), ( d , b ), (d , c), ( d , d )}.
i
] (( A S ] (SA ]). m
A
n
i 1
We give the covering relation and the figure of S by:
(
This shows that
{(d , a ), (d , b), ( d , c )}
m n i 1
i
] (( A S ] (SA ]) is a subsemim
A
n
group of S. We now consider: max{ m , n }
((
A ] (( A S ] (SA i
m
n
]))
m
S
i 1
Let Q {a, d } . For integers m, n > 1, we obtain that Q is an (m, n)-quasi-ideal of S but not a quasi-ideal of S. As in Theorem 1.1, we have the following. Theorem 1.2. Let ( S , , ) be an ordered semigroup. Then the intersection of all (m, n)-quasi-ideals containing a nonempty subset A of S, denoted by [ A]q ,( m , n ) , is an (m, n)-quasiideal containing A of S, and it is of the form
((
max{ m , n }
A ] ( A S ]) S i
m
m
i 1
((
max{ m , n }
A ] ( A S ]) i
m
max{ m , n }
m 1
((
i 1
((
A ] ( A S ]) S i
m
i 1
max{ m , n }
A ] ( A S ]) i
m
m 1
( AS ]
i 1
[ A]q ,( m, n ) (
max{ m , n}
Ai ] (( Am S ] (SAn ]). (1.2)
i 1
((
max{ m , n }
A ] ( A S ]) i
m
max{ m , n }
m2
((
i 1
Proof. Let { Ai | i I } be the set of all (m, n)-quasi-ideals
containing A of S. Then
i I
Ai is a subsemigroup contain-
((
n
i
i
i I
m
((
j
Ai ) S ] (S ( iI Ai ) m
iI
i
i
n
]
iI
and hence
i
iI
i I
Ai . Moreover,,
i I
(
i 1
m
i
A]
m
n
xy ( A S ] ( SA ] ( Let x, y ( i 1
max{ m , n }
i
]
where Q (
max{ m n } i 1
max{ m , n}
A ] (( A S ] ( SA i
m
n
]).
((
k
Ai ] (( Am S ] ( SAn ]), i
A
] (( A S ] (SA ]). Now,, m
n
Ai ] (( Am S ] ( SAn ])]
i 1
i 1
A ; then there exist j, k in {1, 2, ,
max{m , n}} such that x ( A j ] and y ( A ]. If 1 j k
max{m, n} , then
max{ m , n}
( Am S ] ( SAn ]
i 1
( SA ] , then max{ m , n }
n
(
n
n
2
Ai ] (( Am S ] (SAn ])) n ( SAn ].
m
max{ m , n }
m
(Q S ] (SQ ]
(( A S ] ( SA ]) . If x ( A S ] ( SA ] or y ( A S ]
m
( A S]
Then,
n
n
m2
i 1
A ] (( A S ] ( SA ]) . Let x, y ( i 1 m
max{ m , n}
i
i I
Ai is an (m, n)-quasi-ideal of S.
i
m
m
S ((
Next, we will show that (1.2) holds. Clearly, max{ m , n}
i
( A S ]. Similarly,
( A ] (A ] A ( A ] , i I
A ] ( A S ])
n
S ] ( S ( Aj ) ] Aj,
iI
and then
m
i 1
(( A ) S ] (S ( A ) ] (( A ) m
i
i 1
max{ m , n }
ing A of S. For j I , we have
A ] ( A S ])( AS ]
((
max{ m , n}
i 1
Ai ]] ((( Am S ] ( SAn ])]
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(
Ai ] (( Am S ] ( SAn ]).
we let
( B]A { y A∣y b for some b B}.
i 1
Thus
(
max{ m n } i 1
A
i
] (( A S ] (SA ]) m
n
is an (m, n)-quasi-
It is clear that ( B ]A ( B ] , and the equality holds in the following lemma.
ideal containing A of S, and [ A]q ,( m , n )
(
max{ m n } i 1
Ai ] (( A m S ] ( SA n ]).
By max{ m , n}
(
m , n} Ai ] ([ A]q ,( m , n ) [ A]qmax{ ] ,( m , n )
i 1
Lemma 2.1. If A is an (m, n)-ideal of an ordered semigroup ( S , , ) , then ( B ] A ( B ] for any non-empty subset B of A. Lemma 2.2. Let A be an (m, n)-ideal of an ordered semigroup ( S , , ) , and let B A . Then
[ A]q ,( m , n )
(([ BA ]( m , n ) ) m S ([ BA ]( m , n ) ) n ] ( B m SB n ]
and
m n
( A S ] (SA ] (([ A] m
n
) S] m
q ,( m , n )
where [ B A ]( m , n )
( B
i
B m AB n ]A .
i 1
( S ([ A]q ,( m , n ) )
n
] [ A]
q ,( m , n )
,
Proof. We have mn
(( B
it follows that max{ m , n}
(
Ai ] (( Am S ] ( SAn ]) [ A]q ,( m ,n ) .
This shows that (1.2) holds, and the proof is completed. For an element a of an ordered semigroup ( S , , ) , we write [{a}]q ,( m ,n ) (or simply [ a ]q ,( m ,n ) ) by max{ m , n}
B m AB n ]mA S ( B i B m AB n ]nA ]
i 1
i 1
mn
m n
i 1
i 1
mn
mn
i 1
i 1
((( B i B m AB n )m S ( B i B m AB n )n ]A ]
i 1
[ a ]q ,( m , n ) (
mn
i
((( B i B m AB n )m S ( B i B m AB n )n ]]
{a}i ] ((a m S ] (Sa n ]).
mn
m n
i 1
i 1
(( B i B m AB n ) m S ( B i B m AB n )n ].
i 1
In closing this section we quote the following two results proved by Sanborisoot et al. (2012).
m
Lemma 1.1. The following conditions hold for an ordered semigroup ( S , , ) and a S : m m (1) ([ a]( m ,0 ) ) S ( a S ] for any positive integer m. n n (2) S ([a ](0, n ) ) ( Sa ] for any positive integer n. m n m n (3) ([ a]( m , n ) ) S ([ a]( m , n ) ) ( a Sa ] for any positive integers m, n. Theorem 1.3. Let ( S , , ) be an ordered semigroup. Let m, n be positive integers. Let ( m ,0 ) be the set of all ( m, 0) -ideals of S, and let ( 0, n ) be the set of all (0, n)-ideals of S. Then the following conditions hold: (1) S is ( m, 0) -regular if and only if for all R ( m ,0), R (Rm S ] . (2) S is (0, n)-regular if and only if for all L (0, n ) , L ( SLn ] . 2. Main Results Let A be a subsemigroup of an ordered semigroup ( S , , ) . For a non-empty subset B of A,
n
m n Let x (([ BA ]( m , n ) ) S ([ BA ]( m , n ) ) ] . Then x y sz for
some s S and y , z
mn
mn
i 1
i 1
B i B m AB n . If y, z B i ,
p
then y B , z B q for some p, q {1, 2, , m n}; hence m n
i m n x ( B mp SB nq ] ( B m SB n ]. If y B , z B SB , then i 1
mp y B p for some p {1, 2, , m n} ; hence x ( B S mn
i ( B m SB n ) n ] ( B m SB n ]. If y B m SB n , z B , then i 1
z B q for some q {1, 2, , m n}; hence x (( B m SB n ) m SB nq ] ( B m SB n ] . Finally, if y , z B m SB n , then x (( B m SB n ) m S ( B m SB n ) n ] ( B m SB n ] . This shows that
([ B
]
A ( m,n )
) S ([ B m
]
A ( m, n )
)
n
( B m SB n ].
By
( B m SB n ] (([ BA ]( m, n ) ) m S ([ BA ]( m , n ) ) n ], it follows that
(([ BA ]( m , n ) ) m S ([ BA ]( m, n ) ) n ] ( B m SB n ],
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L. Bussaban & T. Changphas / Songklanakarin J. Sci. Technol. 38 (2), 199-206, 2016
as required. This completes the proof.
Theorem 2.1. Let A be an (m, n)-ideal of an ordered semigroup ( S , , ) . Then every (m, n)-ideal of A is an (m, n)ideal of S if and only if for each non-empty subset B of A,
B m SB n [ BA ]( m ,n ) m n
where [ BA ]( m , n )
( B
i
B m AB n ]A .
Proof. Assume first that every (m, n)-ideal of A is an (m, n)ideal of S. Let B A. Since [ B A ]( m ,n ) is an (m, n)-ideal of A, it follows by assumption that [ B A ]( m , n ) is an (m, n)ideal of S. By Lemma 2.2, n
m
n
{( a, b)}
(2.1)
i 1
m
{(a, a ), ( a, b), (b, b), (c, c ), ( d , d )}. We give the covering relation and the figure of S by:
m
n
B SB ( B SB ] (([ BA ]( m , n ) ) S ([ BA ]( m ,n ) ) ] ([ B A ]( m ,n ) ] [ BA ]( m , n ) . Conversely, we assume that the equation (2.1) holds for any non-empty subset of A. Let C be an (m, n)-ideal of A. Then C A and
C m SC n (C C 2 C m n C m AC n ] A (C ] A C . By Lemma 2.1, (C ] C . Therefore, C is an (m, n)-ideal of S. For m = 0, n = 1 (resp. m = 1, n = 0), we have the following corollary: Corollary 2.1. Let A be a left (resp. right) ideal of an ordered semigroup ( S , , ) . Then every left (resp. right) ideal of A is a left (resp. right) ideal of S if and only if for each non-empty subset B of A,
SB ( B AB ] A ( resp., BS ( B BA] A ). Moreover we have the following, taking m = 1, n = 1: Corollary 2.2. Let A be a bi-ideal of an ordered semigroup ( S , , ) . Then every bi-ideal of A is a bi-ideal of S if and only if for each non-empty subset B of A,
BSB ( B B 2 BAB ] A . Example 2.1. Let S {a, b, c, d } be an ordered semigroup with the multiplication and the order relation defined by:
Then A {a, d } is a bi-ideal of S, and {a} is a bi-ideal of A. It is easy to verify that, for each non-empty subset B of A, we have BSB ( B B 2 BAB ] A . Thus, by Corollary 2.2, {a} is a bi-ideal of S. Theorem 2.2. Let Q be an (m, n)-quasi-ideal of an ordered semigroup ( S , , ) . Then every (m, n)-quasi-ideal of Q is an (m, n)-quasi-ideal of S if and only if for each non-empty subset D of Q,
( D m S ] ( SD n ] [ DQ ]q ,( m , n ) where [ DQ ]q ,( m, n ) (
max{m , n }
(2.2)
D i ]Q (( D mQ]Q (QD n ]Q ).
i 1
Proof. Assume that every (m, n)-quasi-ideal of Q is an (m, n)-quasi-ideal of S. If D Q is non-empty, then, by Theorem 1.2, [ DQ ]q ,( m , n ) is an (m, n)-quasi-ideal of Q. By assumption,
( D m S ] ( SD n ] (([ DQ ]q ,( m ,n ) ) m S ] ( S ([ DQ ]q ,( m , n ) ) n ] [ DQ ]q ,( m , n ) . Conversely, we assume that the equation (2.2) holds for any non-empty subset of Q. Let C be an (m, n)-quasiideal of Q. Then C Q and
(C m S ] ( SC n ] [CQ ]q ,( m , n ) C. By Lemma 2.1, (C ] C . Therefore, C is an (m, n)-quasiideal of S. For m = 1, n = 1, we have the following corollary: Corollary 2.3. Let Q be a quasi-ideal of an ordered semigroup ( S , , ) . Then every quasi-ideal of Q is a quasi-ideal of S if and only if for each non-empty subset D of Q,
( DS ] ( SD ] ( D ]Q (( DQ ]Q (QD ]Q ). Example 2.2. Let S {a, b, c, d , f } be an ordered semigroup with the multiplication and the order relation defined by:
204
L. Bussaban & T. Changphas / Songklanakarin J. Sci. Technol. 38 (2), 199-206, 2016 It is easy to verify that Q {a, b, d } is an (m, n)-quasi-ideal of S for any integers m, n 2, and the (m, n)-quasi-ideal of Q is {b, d } . For each non-empty subset C of Q, we have (C m S ] ( SC n ] [CQ ]q ,( m , n ). By Theorem 2.2, {b, d } is also a quasi-ideal of S. Let ( S , , ) be an ordered semigroup, and let m, n be non-negative integers. Then S is said to be (m, n)-regular (Sanborisoot et al., 2012), if for any a in S there exists x in m n m n S such that a a xa , that is, if a ( a Sa ] .
{(a, a), (a, b), (a, c ), (a, d ), ( a, f ), (b, b), (c, c ), ( d , d ), ( f , f )}.
Example 2.4. Let S {a, b, c, d } be an ordered semigroup with the multiplication and the order relation defined by:
We give the covering relation and the figure of S by:
{( a, b), ( a, c ), ( a, d ), ( a, f )}
Then Q {a, c, f } is a quasi-ideal of S, and the quasi-ideals of Q are D1 a , D2 a, c and D3 a, f . For each non-empty subset C of Q, we have (CS ] ( SC ] (C ]Q ((CQ ]Q (QC ]Q ) . By Corollary 2.3, D1 , D2 , D3 are quasiideals of S. Example 2.3. Let S {a, b, c, d , f } be an ordered semigroup with the multiplication and the order relation defined by:
{( a, a ), ( a, b), (a, c), ( a, d ), (b, b), (c, c ), (c, d ), (d , d )}. We give the covering relation and the figure of S by:
{( a, b), (a, c), (c, d )}
Then S is (m, n)-regular for any integer m, n 1 .
{( a, a ), ( a, b), (a, c), ( a, f ), (b, b), (b, c ), (b, f ), (c, c ),
d , d , d , b , d , c , d , f , f , f }. We give the covering relation and the figure of S by:
{( a, b), (b, c), (b, f ), (d , b)}
Theorem 2.3. Let ( S , , ) be an ordered semigroup. Then S is (m, n)-regular if and only if
R ( m ,0 ) , L ( 0, n ) , R L ( R m Ln ]
(2.3)
where ( m ,0) is the set of all (m, 0)-ideals of S and ( 0, n ) is the set of all (0, n)-ideals of S. Proof. The assertion is obvious if m = 0, n = 0. If m = 0, n 0, we have to show that S is (0, n)-regular if and only if L (0 ,n ) , L ( SLn ] , and this follows by Theorem 1.3 (2). Similarly, for m 0, n = 0. This is obtained by Theorem 1.3 (1). Finally, we let m 0, n 0 . Assume that S is (m, n)regular. Let R ( m ,0) and L (0, n ) . We have ( R m Ln ] R L . Let a R L . Since S is regular, there exists x
205
L. Bussaban & T. Changphas / Songklanakarin J. Sci. Technol. 38 (2), 199-206, 2016 m
n
in S such that a a xa . We have m
a a xa
m
n n
xa xa
n
a a
3 m 2
xa xa xa
n
n
nm ( n 1)
Example 2.5. We consider the ordered semigroup which is defined in Example 2.2. We have a 1,1 a , b 1,1 a, b, c (1,1) a, c , d (1,1) a, d , and f (1,1) a, f .Then, by Theorem 2.4, S is regular..
n
L
References
R m Ln ( R m Ln ]. m
n
Thus R L ( R L ]. Conversely, we assume that (2.3) holds. Let a S . Since [ a ]( m ,0) ( m ,0) and S ( 0, n ) , we have
[ a ]( m ,0 ) [a ]( m,0) S (([ a ]( m ,0) ) m S n ] (([ a]( m ,0 ) ) m S ]. m
n
By Lemma 1.1, [ a ]( m ,0) ( a S ] . Similarly, [ a ]( 0, n ) ( Sa ] . From
a [ a ]( m ,0 ) [ a ](0, n )
( a m S ] ( Sa n ] (( a m S ]) m (( Sa n ]) n ( a m S ]( Sa n ] ( a m Sa n ], we conclude that S is (m, n)-regular. We now complete the proof. Corollary 2.4. Let ( S , , ) be an ordered semigroup. Then S is (m, n)-regular if and only if
a S ,[ a ]( m ,0) [ a]( 0,n ) (([a ]( m,0) ) m ([ a](0, n ) ) n ]. Theorem 2.4. Let ( S , , ) be an ordered semigroup. Then S is (m, n)-regular if and only if
a S ,[ a ]( m , n ) ( a m Sa n ]. Proof. Assume that S is (m, n)-regular. Let a S and x [ a ]( m , n ) . Then, by Theorem 1.1, x y for some y in m n
i 1
m
n
n
a nm ( n 1) ( xa n ) n
i
n
Conversely, if a S , then a [a ]( m ,n ) ( a Sa ], and hence S is (m, n)-regular.
a
m
m
2 m 1
R
n
Since ( a Sa ] [ a]( m , n ) ,[a ]( m, n ) (a Sa ].
n
a Sa . If y a m Sa n , we are done. Suppose that
mn
y a i ; then y a p for some p {1, 2, , m n} . We i 1
have
x ( a p ] (( a m Sa n ] p ] (( a m Sa n ]] ( a m Sa n ].
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Yousafzai, F., Khan, W., Guo, W. and Khan, M. 2014. On (m, n)-ideals of left almost semigroups. Journal of Semigroup Theory and Applications. 2014, Article ID.1
