ordered $\\mathcal{la}$-semigroups in terms of interval valued fuzzy ideals

$: Ordered $\\mathcal{LA}$-Semigroups in Terms of Interval Valued Fuzzy Ideals$
Journal of Advanced Research inPure MathematicsOnline ISSN: 1943-2380

Vol. 5, Issue. 1, 2013, pp. 100-117doi: 10.5373/jarpm.1240.122111

Ordered LA-semigroups in terms of interval valuedfuzzy ideals

Asghar Khan1,∗, Faisal1, Waqar Khan1, Naveed Yaqoob2

1 Department of Mathematics, COMSATS Institute of Information Technology,Abbottabad, Khyber Pukhtoon Khwa, Pakistan.2 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.

Abstract. In this paper, we have given the concept of interval valued fuzzy orderedLA-semigroups and characterized an intra-regular ordered LA-semigroup in terms ofinterval valued fuzzy left (right, two-sided) ideals.

Keywords: Ordered LA-semigroup; Intra-regular ordered LA-semigroup; Left invertive law;Interval valued fuzzy two-sided ideal.

Mathematics Subject Classification 2010: 20M10, 20N99.

1 Introduction

Given a set S, a fuzzy subset of S is an arbitrary mapping f : S → [0, 1], where [0, 1] isthe unit segment of a real line. This fundamental concept of fuzzy set was first given byZadeh [20] in 1965. Fuzzy groups have been first considered by Rosenfeld [14] and fuzzysemigroups by Kuroki [7]. Also see [8, 9].

There are many kinds of extensions in the fuzzy set theory, like, intuitionistic fuzzysets, i-v fuzzy sets, vague sets. Zadeh [21] made an extension of the concept of a fuzzysubset by an interval valued fuzzy subset, i.e., a fuzzy subset with an interval valuedmembership function. In [21], Zadeh also constructed a method of approximate inferenceusing his interval valued fuzzy subsets. Yaqoob et. al. [17] characterized left almostsemigroups by their interval valued fuzzy ideals, also see [3,18,19]. Biswas [1] defined theinterval valued fuzzy subgroups of Rosenfeld’s nature, and investigated some elementaryproperties. In [12], Narayanan and Manikantan, introduced the concept of an intervalvalued fuzzy left (right, two-sided, interior, bi-) ideal generated by an interval valued

∗Correspondence to: Asghar Khan, Department of Mathematics, COMSATS Institute of InformationTechnology, Abbottabad, K.P.K, Pakistan. Email: [email protected]†Received: 21 December 2011, revised: 5 April 2012, accepted: 26 June 2012.

http://www.i-asr.com/Journals/jarpm/ 100 c⃝2013 Institute of Advanced Scientific Research

A. Khan, Faisal, W. Khan and N. Yaqoob 101

fuzzy subset in semigroups and also they presented characterizations of such generatedinterval valued fuzzy ideals.

The concept of a Left Almost semigroup (LA-semigroup) [4] was first given by M.A. Kazim and M. Naseeruddin in 1972. An LA-semigroup is a groupoid having the leftinvertive law

(ab)c = (cb)a, for all a, b, c ∈ S. (1)

An LA-semigroup is a non-associative and non-commutative algebraic structure midway between a groupoid and a commutative semigroup, nevertheless, it posses manyinteresting properties which we usually find in associative and commutative algebraicstructures. The left identity in an LA-semigroup if exists is unique [10].

In an LA-semigroup, the medial law [4] holds

(ab)(cd) = (ac)(bd), for all a, b, c, d ∈ S. (2)

In an LA-semigroup S with left identity, the paramedial law [10] holds

(ab)(cd) = (dc)(ba), for all a, b, c, d ∈ S. (3)

If an LA-semigroup contains a left identity, then by using medial law, the followinglaw [10] holds

a(bc) = b(ac), for all a, b, c ∈ S. (4)

An LA-semigroup with right identity becomes a commutative monoid [10]. The connec-tion of a commutative inverse semigroup with an LA-semigroup has been given in [11]as, a commutative inverse semigroup (S, ) becomes an LA-semigroup (S, ·) undera · b = b a−1, for all a, b ∈ S . An LA-semigroup S with left identity becomes a semi-group (S, ) defined as, for all x, y ∈ S, there exists a ∈ S such that xy = (xa)y [15]. AnLA-semigroup is the generalization of a semigroup theory [10] and has vast applicationsin collaboration with semigroup like other branches of mathematics. An LA-semigrouphas wide range of applications in theory of flocks [13]. The connection of LA-semigroupswith the vector spaces over finite fields has been investigated in [5].

From the above discussion, we see that LA-semigroups have very closed link withsemigroups and vector spaces which shows the importance and applications of LA-semigroups.

The concept of an ordered LA-semigroup was first given by Khan and Faisal in [6]which is the generalization of an ordered semigroup. This paper concerned the relation-ship between interval valued fuzzy sets and an ordered LA-semigroup.

Definition 1.1. An ordered LA-semigroup (po-LA-semigroup) is a structure (S, .,≤)in which the following conditions hold [6].

(i) (S, .) is an LA-semigroup.

(ii) (S,≤) is a poset (reflexive, anti-symmetric and transitive).

(iii) For all a, b and x ∈ S, a ≤ b implies ax ≤ bx and xa ≤ xb.

102 Ordered LA-semigroups in terms of interval valued fuzzy ideals

The following example is a natural example of an ordered LA-semigroup called areal ordered LA-semigroup [6].

Example 1.1. [2] Consider an open interval RO = (0, 1) of real numbers under thebinary operation of multiplication. Define a ∗ b = ba−1r−1, for all a, b, r ∈ RO, then(RO, ∗,≤) is an ordered LA-semigroup under the usual order ”≤”.

2 Preliminaries

An interval number is a = [a−, a+], where 0 6 a− 6 a+ 6 1. Let D[0, 1] denote thefamily of all closed subintervals of [0, 1], i.e.,

D[0, 1] = a = [a−, a+] : a− 6 a+, for a−, a+ ∈ [0, 1].

Where the elements in D[0, 1] are called the interval numbers on [0, 1], 0 = [0, 0] and1 = [1, 1].

We define the operations 6, =, <, min and max in case of two elements in D[0, 1].We consider two elements a = [a−, a+] and b = [b−, b+] in D[0, 1]. Then

(C1) a 6 b if and only if a− 6 b− and a+ 6 b+,(C2) a = b if and only if a− = b− and a+ = b+,(C3) a < b if and only if a− < b−, a+ < b+ and a = b,(C4) mina, b = a ∧ b = [a− ∧ b−, a+ ∧ b+],(C5) maxa, b = a ∨ b = [a− ∨ b−, a+ ∨ b+.Let X be a crisp set, then the mapping µ : X → D[0, 1] is called an interval valued

fuzzy set (for short, i-v fuzzy set) onX. Let IV F (X) denote the family of the interval val-ued fuzzy sets onX. For each µ ∈ IV F (X), we have µ(x) = ⟨x, [µ−(x), µ+(x)]⟩ : x ∈ X ,where µ−(x) 6 µ+(x), for all x ∈ X. Then the ordinary fuzzy sets µ− : X → [0, 1] andµ+ : X → [0, 1] are called a lower fuzzy set and an upper fuzzy set of µ, respectively.

In addition, we define ∅(x) = [0, 0] and X(x) = [1, 1], for all x ∈ X. Obviously,

∅, X ∈ IV F (X). Let µ, λ ∈ IV F (X) and x ∈ X, we define(µ ∪ λ)(x) = maxµ(x), λ(x) = µ(x) ∨ λ(x);(µ ∩ λ)(x) = minµ(x), λ(x) = µ(x) ∧ λ(x).

Definition 2.1. An i-v fuzzy subset µ of an ordered LA-semigroup S is called an i-vfuzzy left (resp. right) ideal of S if µ satisfies the following

x ≤ y ⇒ µ(x) ≥ µ(y), for all x, y ∈ S,

andµ (xy) ≥ µ (y) (resp. µ (xy) ≥ µ (x)), for all x, y ∈ S.

An i-v fuzzy subset µ of an ordered LA-semigroup S is called an i-v fuzzy two-sidedideal or an i-v fuzzy ideal of S if it is both an i-v fuzzy left ideal and an i-v fuzzy rightideal of S.


Definition 2.2. An i-v fuzzy subset µ of an ordered LA-semigroup is called i-v fuzzysemiprime if µ(a) ≥ µ(a2), for all a in S.

Definition 2.3. An i-v fuzzy subset µ of an ordered LA-semigroup is said to be idem-potent if µ µ = µ.

Definition 2.4. A subset A of an ordered LA-semigroup S is called semiprime if a2 ∈ Aimplies a ∈ A.

Definition 2.5. A non-empty subset A of an S is called a left (right) ideal of S if

(i) SA ⊆ A (AS ⊆ A).

(ii) If a ∈ A and b is in S such that b ≤ a, then b is in A.

Definition 2.6. A subset A of S is called a two-sided ideal of S if it is both a left anda right ideal of S.

Let x ∈ S, then Ax = (y, z) ∈ S × S\x ≤ yz .Let µ and γ be two i-v fuzzy subsets of an ordered LA-semigroup S then µ γ is

defined as

(µ γ) (x) =

∨x≤yz

µ (y) ∧ γ (z) if x ≤ yz, for some y, z ∈ S

[0, 0] otherwise.

Let S be an an ordered LA-semigroup and let ∅ = W ⊆ S, then the interval valuedcharacteristic function ΥW of W is defined as

ΥW (x) =

[1, 1], if x ∈ W[0, 0], if x /∈ W.

3 Interval valued fuzzy sets in ordered LA-semigroups

Let IV F (S) denote the set of all i-v fuzzy subsets of an ordered LA-semigroup.

Theorem 3.1. Let S be an ordered LA-semigroup, then the set (IV F (S), , ⊆) is anordered LA-semigroup.

Proof. Clearly IV F (S) is closed. Let µ, γ and α be in IV F (S) and let x be any elementof S such that it is not expressible as product of two elements in S. Then we have,

((µ γ) α) (x) = [0, 0] = ((α γ) µ) (x).

Let Ax = ∅, then there exist y and z in S such that (y, z) ∈ Ax. Therefore by using (1),


we have

((µ γ) α)(x) =∨

(y,z)∈Ax

(µ γ) (y) ∧ α(z)

=∨

(y,z)∈Ax

∨(p,q)∈Ay

µ(p) ∧ γ(q) ∧ α(z)

=

∨x≤(pq)z

µ(p) ∧ γ(q) ∧ α(z)

=∨

x≤(zq)p

α(z) ∧ γ(q) ∧ µ(p)

=∨

(w,p)∈Ax

∨(z,q)∈Aw

(α(z) ∧ γ(q) ∧ µ(p))

=

∨(w,p)∈Ax

(α γ) (w) ∧ f(p) = ((α γ) µ)(x).

Hence (IV F (S), ) is an LA-semigroup.Assume that µ ⊆ γ and let Ax = ∅ for any x ∈ S, then

(µ α)(x) = [0, 0] = (γ α)(x) =⇒ µ α ⊆ γ α.

Similarly we can show that µ α ⊇ γ α. Let Ax = ∅, then there exist y and z in S suchthat (y, z) ∈ Ax, therefore

(µ α)(x) =∨

(y,z)∈Ax

µ (y) ∧ α(z) ≤∨

(y,z)∈Ax

γ (y) ∧ α(z) = (γ α)(x),

Similarly we can show that µ α ⊇ γ α. It is easy to see that IV F (S) is a poset. Thus(IV F (S), , ⊆) is an ordered LA-semigroup.

Corollary 3.2. The left invertive law holds in IV F (S).

Corollary 3.3. The medial law holds in IV F (S).

Corollary 3.4. For any fuzzy subsets µ, γ and α of IV F (S), the following conditionsare equivalent:

(i) (µ γ) α = γ (µ α),(ii) (µ γ) α = γ (α µ).

Note that the Corollaries 3.2 and 3.3 are the analogues definitions of (1) and (2) infuzzy context.

Theorem 3.5. If an ordered LA-semigroup S has a left identity, then the followingproperties hold in IV F (S).

(i) (µ γ) (α β) = (β α) (γ µ) , for all µ, γ, α and β in IV F (S).(ii) µ (γ α) = γ (µ α) , for all µ, γ and α in IV F (S).


Proof. (i) : IfAx = ∅ for x ∈ S, then((µ γ) (α β)

)(x) = [0, 0] =

((β α) (γ µ)

)(x).

Let Ax = ∅, then there exist y and z in S such that (y, z) ∈ Ax. Therefore by using (3),we have((µ γ) (α β)

)(x) =

∨(y,z)∈Ax

(µ γ) (y) ∧ (α β)(z)

=∨

(y,z)∈Ax

∨(p,q)∈Ay

µ(p) ∧ γ(q) ∧∨

(u,v)∈Az

α(u) ∧ β(v)

=

∨x≤(pq)(uv)

µ(p) ∧ γ(q) ∧ α(u) ∧ β(v)

=

∨x≤(vu)(qp)

β(v) ∧ α(u) ∧ γ(q) ∧ µ(p)

=∨

(m,n)∈Ax

∨(v,u)∈Am

β(v) ∧ α(u)

∧

∨(q,p)∈An

γ(q) ∧ µ(p)

=

∨(m,n)∈Ax

(β α)(m) ∧ (γ µ)(n)

=

((β α) (γ µ)

)(x).

Thus, (µ γ) (α β) = (β α) (γ µ) , for all x in S.(ii) : Let x be an arbitrary element of S. If Ax = ∅ for x ∈ S, then (µ (γ α)) (x) =

[0, 0] = (γ (µ α)) (x). Let Ax = ∅, then there exist y and z in S such that (y, z) ∈ Ax.Now by using (4), we have

(µ (γ α)) (x) =∨

(y,z)∈Ax

µ (y) ∧ (γ α) (z)

=∨

(y,z)∈Ax

µ (y) ∧∨

(p,q)∈Az

γ(p) ∧ α(q)

=

∨x≤y(pq)

µ (y) ∧ γ(p) ∧ α(q)

=∨

x≤p(yq)

γ (p) ∧ µ(y) ∧ α(q)

=∨

(p,w)∈Ax

γ (p) ∧∨

(y,q)∈Aw

µ(y) ∧ α(q)

=

∨(p,w)∈Ax

γ (p) ∧ (µ α) (w) = (γ (µ α)) (x).

Thus, (µ (γ α)) (x) = (γ (µ α)) (x), for all x in S.


Note that Theorem 3.5 gives the analogues definitions of (3) and (4) in fuzzy context.

Proposition 3.6. An ordered LA-semigroup S with IV F (S) = (IV F (S))2 is a com-mutative ordered semigroup if and only if (µ γ) α = µ (α γ) holds for all intervalvalued fuzzy subsets µ, γ, α ∈ IV F (S).

Proof. Let an ordered LA-semigroup S be a commutative ordered semigroup. For anyi-v fuzzy subsets µ, γ,α ∈ IV F (S), if Ax = ∅ for any x ∈ S, then ((µ γ) α)(x) =[0, 0] = (µ (α γ))(x). Let Ax = ∅, then there exist s and t in S such that (s, t) ∈ Ax,therefore by use of (1) and commutative law, we get

((µ γ) α)(x) =∨

(s,t)∈Ax

(µ γ)(s) ∧ α(t)

=∨

(s,t)∈Ax

∨(m,n)∈As

µ(m) ∧ γ(n) ∧ α(t)

=

∨x≤(mn)t

µ(m) ∧ α(t) ∧ γ(n)

=∨

x≤(tn)m

µ(m) ∧ α(t) ∧ γ(n)

=∨

x≤m(tn)

µ(m) ∧ α(t) ∧ γ(n)

=∨

(m,p)∈Ax

∨(t,n)∈Ap

µ(m) ∧ α(t) ∧ γ(n)

=

∨(m,p)∈Ax

µ(m) ∧ (α γ)(p) = (µ (α γ))(x).

Conversely, let (µ γ) α = µ(α γ) holds for all i-v fuzzy subsets µ, γ,α ∈ IV F (S). Wehave to show that S is a commutative ordered semigroup. Let µ and γ be any arbitraryi-v fuzzy subsets of S, if Ax = ∅ for any x ∈ S, then (µ γ)(x) = [0, 0] = (γ µ)(x). LetAx = ∅, then there exist s and t in S such that (s, t) ∈ Ax. Since IV F (S) = (IV F (S))2,


so µ = α β, where α and β are any i-v fuzzy subsets of S. Now by (1), we have

(µ γ)(x) = ((α β) γ)(x) =∨

(s,t)∈Ax

(α β)(s) ∧ γ(t)

=∨

(s,t)∈Ax

∨(m,n)∈As

α(m) ∧ β(n) ∧ γ(t)

=

∨x≤(mn)t

γ(t) ∧ β(n) ∧ α(m)

=

∨x≤(tn)m

γ(t) ∧ β(n) ∧ α(m)

=∨

(p,m)∈Ax

∨(t,n)∈Ap

γ(t) ∧ β(n) ∧ α(m)

=

∨(p,m)∈Ax

(γ β)(p) ∧ α(m)

= (γ (α β))(x).

This shows that µ γ = γ (α β) = γ µ. Therefore commutative law holds in S.Now if Ax = ∅ for any x ∈ S, then ((µ γ) β)(x) = [0, 0] = (µ (γ β))(x). Let

Ax = ∅, then there exist s and t in S such that (s, t) ∈ Ax, therefore by use of (1) andcommutative law, we get

((µ γ) β)(x) =∨

(s,t)∈Ax

(µ γ)(s) ∧ β(t)

=∨

(s,t)∈Ax

∨(m,n)∈As

µ(m) ∧ γ(n) ∧ β(t)

=

∨x≤(mn)t

µ(m) ∧ γ(n) ∧ β(t)

=

∨x≤(tn)m

µ(m) ∧ γ(n) ∧ β(t)

=

∨x≤m(tn)

µ(m) ∧ γ(n) ∧ β(t)

=

∨x≤m(nt)

µ(m) ∧ γ(n) ∧ β(t)

=∨

(m,p)∈Ax

∨(n,t)∈Ap

µ(m) ∧ γ(n) ∧ β(t)

=

∨(m,p)∈Ax

µ(m) ∧ (γ β)(p)

= (µ (γ β)(x).


Theorem 3.7. CM = µ | µ ∈ IV F (S), µ α = µ, where α = α α is a commutativemonoid in S.

Proof. The fuzzy subset CM of S is non-empty since α α = α, which implies that αis in CM. Let µ and γ be i-v fuzzy subsets of S in CM, then µ α = µ and γ α = γ.If Ax = ∅ for x ∈ S, then (µ γ) (x) = [0, 0] = ((µ γ) α) (x). Let Ax = ∅, then thereexist y and z in S such that (y, z) ∈ Ax. Therefore by using (2), we have

(µ γ)(x) =∨

(y,z)∈Ax

(µ α) (y) ∧ (γ α) (z)

=∨

(y,z)∈Ax

∨(p,q)∈Ay

µ(p) ∧ α(q) ∧∨

(u,v)∈Az

γ(u) ∧ α(v)

=

∨x≤(pq)(uv)

µ(p) ∧ α(q) ∧ γ(u) ∧ α(v)

=∨

x≤(pu)(qv)

µ(p) ∧ γ(u) ∧ α(q) ∧ α(v)

=∨

(m,n)∈Ax

∨(p,u)∈Am

µ(p) ∧ γ(u) ∧∨

(q,v)∈An

α(q) ∧ α(v)

=

∨(m,n)∈Ax

(µ γ)(m) ∧ (α α)(n) = ((µ γ) (α α))(x).

Thus µ γ = (µ γ) (α α) = (µ γ) α, which implies that CM is closed.Now if Ax = ∅ for x ∈ S, then (µ γ) (x) = [0, 0] = (γ µ) (x). Let Ax = ∅, then

there exist y and z in S such that (y, z) ∈ Ax. Therefore by using (1), we have

(µ γ)(x) =∨

(y,z)∈Ax

(µ α) (y) ∧ γ(z)

=∨

(y,z)∈Ax

∨(p,q)∈Ay

µ (p) ∧ α(q) ∧ γ(z)

=

∨x≤(pq)z

γ(z) ∧ α(q) ∧ µ (p)

=∨

x≤(zq)p

γ(z) ∧ α(q) ∧ µ (p)

=∨

(t,p)∈Ax

∨(z,q)∈At

γ(z) ∧ α(q) ∧ µ (p)

=

∨(t,p)∈Ax

(γ α)(t) ∧ µ (p) = ((γ α) µ)(x).


Thus µ γ = (γ α) µ = γ µ, which implies that commutative law holds in CM andassociative law holds in CM due to commutativity. Since for any fuzzy subset µ in CM,we have µ α = µ (where α is fixed) implies that α is a right identity in S and hence anidentity.

Note that an ordered LA-semigroup S can be considered an i-v fuzzy subset of itselfand we write S(x) = [1, 1], for all x in S.

Lemma 3.8. In an ordered LA-semigroup S with left identity, we have S S = S.

Proof. The proof of this lemma is straightforward.

4 Interval valued fuzzy ideals in intra-regular ordered LA-semigroups

An element a of an ordered LA-semigroup S is called an intra-regular element of Sif there exist x, y ∈ S such that a ≤ (xa2)y [6] and S is called an intra-regular ifevery element of S is an intra-regular or equivalently, A ⊆ ((SA2)S] for all A ⊆ S anda ∈ ((Sa2)S], for all a ∈ S.

Example 4.1. Let S = 1, 2, 3, 4, 5 be an ordered LA-semigroup with left identity 4with the following multiplication table and order below.

. 1 2 3 4 5

1 1 1 1 1 12 1 2 2 2 23 1 2 4 5 34 1 2 3 4 55 1 2 5 3 4

≤:= (1, 1), (2, 1), (3, 3), (4, 4), (5, 5), (2, 2)

It is easy to see that S is an intra-regular. We define

µ =⟨

x, [µ−(x), µ+(x)]⟩: x ∈ X

= ⟨1, [0.3, 0.4]⟩ , ⟨2, [0.5, 0.6]⟩ , ⟨3, [0.1, 0.2]⟩ , ⟨4, [0.1, 0.2]⟩ , ⟨5, [0.1, 0.2]⟩ .

Clearly µ is an i-v fuzzy two-sided ideal of S.

Lemma 4.1. For any subset A of an ordered LA-semigroup S, the following propertiesholds.

(i) A is an ordered LA-subsemigroup of S if and only if ΥA is an i-v fuzzy orderedLA-subsemigroup of S.

(ii) A is left (right, two-sided) ideal of S if and only if ΥA is an i-v fuzzy left (right,two-sided) ideal of S.

(iii) For any non-empty subsets A and B of an ordered LA-semigroup S, ΥA ΥB =Υ(AB] holds.


Proof. It is straightforward.

Lemma 4.2. Every right (left, two-sided) ideal of an ordered LA-semigroup S issemiprime if and only if their characteristic functions are i-v fuzzy semiprime.

Proof. Let R be any right ideal of an ordered LA-semigroup S, then by Lemma 4.1, thei-v characteristic function of R, that is, ΥR is an i-v fuzzy right ideal of S. Let a2 ∈ R,then ΥR(a

2) = [1, 1] and assume that R is semiprime, then a ∈ R, which implies thatΥR(a) = [1, 1]. Thus we get ΥR(a

2) = ΥR(a), therefore ΥR is an i-v fuzzy semiprime.The converse is simple. The same holds for left and two-sided ideal of S.

Corollary 4.3. Let S be an ordered LA-semigroup, then every right (left, two-sided)ideal of S is semiprime if every i-v fuzzy right (left, two-sided) ideal of S is an i-v fuzzysemiprime.

Lemma 4.4. Every i-v fuzzy right ideal of an ordered LA-semigroup S with left identityis an i-v fuzzy left ideal of S.

Proof. Assume that S is an ordered LA-semigroup with left identity and let µ be an i-vfuzzy right ideal of S, then by using (1), we have

µ(ab) = µ((ea)b) = µ((ba)e) ≥ µ(b),

which show that µ is an i-v fuzzy left ideal of S.

The converse of above is not true in general. For this, let us consider an orderedLA-semigroup S with left identity d defined in the following multiplication table andorder below.

· a b c d e

a a a a a ab a e e c ec a e e b ed a b c d ee a e e e e

≤:= (a, a), (b, a), (c, c), (d, d), (e, e), (b, b).

We define

µ =⟨

x, [µ−(x), µ+(x)]⟩: x ∈ X

= ⟨a, [0.1, 0.8]⟩ , ⟨b, [0.7, 0.8]⟩ , ⟨c, [0.4, 0.6]⟩ , ⟨d, [0.3, 0.8]⟩ , ⟨e, [0.3, 0.6]⟩ .

It is easy to observe that µ is an i-v fuzzy left ideal of S but it is not an i-v fuzzy rightideal of S, because µ(bd) µ(b).

The proof of following Lemma is same as in [16].


Lemma 4.5. In S, the following are true.

(i) A ⊆ (A] for all A ⊆ S.

(ii) If A ⊆ B ⊆ S, then (A] ⊆ (B] .

(iii) (A] (B] ⊆ (AB] for all A,B ⊆ S.

(iv) (A] = ((A]] for all A ⊆ S.

(vi) ((A] (B]] = (AB] for all A,B ⊆ S.

Lemma 4.6. Let µ be an i-v fuzzy subset of an intra-regular ordered LA-semigroup Swith left identity, then µ is an i-v fuzzy left ideal of S if and only if µ is an i-v fuzzyright ideal of S.

Proof. Assume that S is an intra-regular ordered LA-semigroup with left identity andlet µ be an i-v fuzzy left ideal of S. Now for a, b ∈ S there exist x, y, x

′, y

′ ∈ S such thata ≤ (xa2)y and b ≤ (x

′b2)y

′. Now by using (1), (3) and (4), we have

µ(ab) ≥ µ(((xa2)y)b) = µ((by)(x(aa))) = µ(((aa)x)(yb))

= µ(((xa)a)(yb)) = µ(((xa)(ea))(yb)) = µ(((ae)(ax))(yb))

= µ((a((ae)x))(yb)) = µ(((yb)((ae)x))a) ≥ µ(a),

which implies that µ is an i-v fuzzy right ideal of S.

Conversely let µ be an i-v fuzzy right ideal of S. Now by using (4) and (3), we have

µ(ab) ≥ µ(a((x′b2)y

′) = µ((x

′b2)(ay

′)) = µ((y

′a)(b2x

′))

= µ(b2((y′a)x)) ≥ µ(b),

which implies that µ is an i-v fuzzy left ideal of S.

Note that an i-v fuzzy left ideal and an i-v fuzzy right ideal coincide in an intra-regularordered LA-semigroup S with left identity.

Lemma 4.7. Every i-v fuzzy two-sided ideal of an intra-regular ordered LA-semigroupS with left identity is an i-v fuzzy semiprime.

Proof. Assume that µ is an i-v fuzzy two-sided ideal of an intra-regular ordered LA-semigroup S with left identity and let a ∈ S, then there exist x, y ∈ S such thata ≤ (xa2)y. Now by using (3) and (4), we have

µ(a) ≥ µ((xa2)y) = µ((xa2)(ey)) = µ((ye)(a2x)) = µ(a2((ye)x)) ≥ µ(a2).

Thus µ is an i-v fuzzy semiprime.

Theorem 4.8. Let S be an ordered LA-semigroup with left identity, then the followingstatements are equivalent.

(i) S is an intra-regular.

(ii) Every i-v fuzzy two-sided ideal of S is an i-v fuzzy semiprime.


Proof. (i) =⇒ (ii) can be followed by Lemma 4.7.

(ii) =⇒ (i) : Let S be an ordered LA-semigroup with left identity and let every i-vfuzzy two-sided ideal of S is an i-v fuzzy semiprime. Since (a2S] is a two-sided idealof S [6], therefore by using Corollary 4.3, (a2S] is semiprime. Clearly a2 ∈ (a2S] [6],therefore a ∈ (a2S]. Now by using (1), we have

a ∈ (a2S] = ((aa)S] = ((Sa)a] ⊆ ((Sa)(a2S)] = (((a2S)a)S]

= (((aS)a2)S] = ((Sa2)(aS)] ⊆ ((Sa2)S].

Which shows that S is an intra-regular.

Lemma 4.9. Let S be an ordered LA-semigroup, then the following holds.

(i) An i-v fuzzy subset µ is an i-v fuzzy ordered LA-subsemigroup of S if and onlyif µ µ ⊆ µ.

(ii) An i-v fuzzy subset µ is i-v fuzzy left (right) ideal of S if and only if S µ ⊆ µ(µ S ⊆ µ).

Proof. The proof of this lemma is straightforward.

Theorem 4.10. For an ordered LA-semigroup S with left identity, the following con-ditions are equivalent.

(i) S is an intra-regular.

(ii) R ∩ L = (RL], R is any right ideal and L is any left ideal of S such that R issemiprime.

(iii) µ ∩ γ = µ γ, µ is any i-v fuzzy right ideal and γ is any i-v fuzzy left ideal of Ssuch that µ is an i-v fuzzy semiprime.

Proof. (i) =⇒ (iii) : Assume that S is an intra-regular ordered LA-semigroup. Let µbe any i-v fuzzy right ideal and γ be any i-v fuzzy left ideal of S. Now for a ∈ S thereexist x, y ∈ S such that a ≤ (xa2)y. Now by using (4), (1) and (3), we have

a ≤ (x(aa))y = (a(xa))y = (y(xa))a ≤ (y(x((xa2)y)))a = (y((xa2)(xy)))a

= (y((yx)(a2x)))a = (y(a2((yx)x)))a = (a2(y((yx)x)))a.

Therefore

(µ γ)(a) =∨

a≤(a2(y((yx)x)))a

µ(a2(y((yx)x))) ∧ γ(a)

≥ µ(a) ∧ γ(a) = (µ ∩ γ)(a),

which imply that µ γ ⊇ µ ∩ γ and by using Lemma 4.9, µ γ ⊆ µ ∩ γ, thereforeµ ∩ γ = µ γ.

(iii) =⇒ (ii) : Let R be any right ideal and L be any left ideal of an ordered LA-semigroup S, then by Lemma 4.1, ΥR and ΥL are i-v fuzzy right and i-v fuzzy left ideals


of S respectively. As (RL] ⊆ R∩L is obvious [6]. Let a ∈ R∩L, then a ∈ R and a ∈ L.Now by using Lemma 4.1 and given assumption, we have

Υ(RL]

(a) = (ΥR ΥL)(a) = (ΥR ∩ΥL)(a) = ΥR(a) ∧ΥL(a) = [1, 1],

which imply that a ∈ (RL] and therefore R∩L = (RL]. Now by using Corollary 4.3,R is semiprime.

(ii) =⇒ (i) : Let S be an ordered LA-semigroup with left identity, then clearly (Sa]is a left ideal of S [6] such that a ∈ (Sa] and (a2S] is a right ideal of S such thata2 ∈ (a2S]. Since by assumption, (a2S] is semiprime, therefore a ∈ (a2S]. Now by usingLemma 4.5, (3), (1) and (4), we have

a ∈ (a2S] ∩ (Sa] = ((a2S](Sa]] = (a2S](Sa] ⊆ ((a2S)(Sa)] = ((aS)(Sa2)]

= (((Sa2)S)a] = (((Sa2)(eS))a] ⊆ (((Sa2)(SS))a] = (((SS)(a2S))a]

= ((a2((SS)S))a] ⊆ ((a2S)S] = ((SS)(aa)] = ((aa)(SS)] ⊆ ((aa)S]

= ((Sa)a] ⊆ ((Sa)(a2S)] = (((a2S)a)S] = (((aS)a2)S] ⊆ ((Sa2)S].


Theorem 4.11. An ordered LA-semigroup S with left identity is an intra-regular ifand only if for each i-v fuzzy two-sided ideal µ of S, µ(a) = µ(a2), for all a in S.

Proof. Assume that S is an intra-regular ordered LA-semigroup with left identity andlet µ be an i-v fuzzy two-sided ideal of S. Let a ∈ S, then there exist x, y ∈ S such thata ≤ (xa2)y. Now by using (3) and (4), we have

µ(a) ≥ µ((xa2)y) = µ((xa2)(ey)) = µ((ye)(a2x)) = µ(a2((ye)x))

≥ µ(a2) = µ(aa) ≥ µ(a) =⇒ µ(a) = µ(a2).

Thus µ(a) = µ(a2) holds for all a in S.Conversely, assume that for any i-v fuzzy two-sided ideal µ of S, µ(a) = µ(a2) holds

for all a in S. As (a2S] is a two-sided ideal of S with left identity, then by Lemma4.1, Υ(a2S] is an i-v fuzzy two-sided ideal of S. Therefore by given assumption andusing the fact that a2 ∈ (a2S], we have Υ(a2S](a) = Υ(a2S](a

2) = [1, 1], which impliesthat a ∈ (a2S]. Now by using (4) and (2), we have a ∈ ((Sa2)S] and therefore S is anintra-regular.

Lemma 4.12. Every two-sided ideal of an ordered LA-semigroup S with left identityis semiprime.

Proof. It is straightforward.

Theorem 4.13. Let S be an ordered LA-semigroup with left identity, then the followingconditions are equivalent.

(i) S is an intra-regular.(ii) Every i-v fuzzy two-sided ideal of S is idempotent.


Proof. (i) =⇒ (ii) : Assume that S is an intra-regular ordered LA-semigroup with leftidentity and let a ∈ S, then there exist x, y ∈ S such that a ≤ (xa2)y. Now by using(4), (1) and (3), we have

a ≤ (x(aa))y = (a(xa))y = (y(xa))a = ((ex)(ya))a = ((ay)(xe))a = (((xe)y)a)a.

Let µ be an i-v fuzzy two-sided ideal of S, then by using Lemma 4.1, we have µ µ ⊆ µand also we have

(µ µ)(a) =∨

a≤(((xe)y)a)a

µ(((xe)y)a) ∧ µ(a) ≥ µ(a) ∧ µ(a) = µ(a),

which implies that µ µ ⊇ µ. Now by using Lemma 4.1, µ µ ⊆ µ and therefore µis idempotent.

(ii) =⇒ (i) : Assume that every i-v fuzzy two-sided ideal of an ordered LA-semigroupS with left identity is idempotent and let a ∈ S. Since (a2S] is a two-sided ideal of S,therefore by Lemma 4.1, its characteristic function Υ(a2S] is an i-v fuzzy two-sided idealof S. Since a2 ∈ (a2S] so by Lemma 4.12 a ∈ (a2S] and therefore Υ(a2S] = [1, 1]. Nowby using the given assumption and Lemma 4.1, we have

Υ(a2S] Υ(a2S] = µ+Ω(a2S]

and Υ(a2S] Υ(a2S] = Υ((a2S]]2

.

Thus, we have Υ((a2S]]2

(a) = µ+Ω(a2S]

(a) = [1, 1], which imply that a ∈ ((a2S]]2. Now

by using Lemma 4.5 and (3), we have

a ∈ ((a2S]]2 = (a2S]2 = (a2S](a2S] ⊆ ((a2S)(a2S)] = ((Sa2)(Sa2)] ⊆ ((Sa2)S].


Theorem 4.14. For an ordered LA-semigroup S with left identity, the following con-ditions are equivalent.

(i) S is an intra-regular.(ii) µ = (S µ)2, where µ is any i-v fuzzy two-sided ideal of S.

Proof. (i) =⇒ (ii) : Let S be a an intra-regular ordered LA-semigroup and let µ beany i-v fuzzy two-sided ideal of S, then it is easy to see that S µ is also an i-v fuzzytwo-sided ideal of S. Now by using Theorem 4.13, S µ is idempotent and therefore, wehave

(S µ)2 = S µ ⊆ µ.

Now let a ∈ S, since S is an intra-regular therefore there exists x ∈ S such thata ≤ (xa2)y. Now by using (4), (3) and (1), we have

a ≤ (x(aa))y = (a(xa))y ≤ (((xa2)y)(xa))(ey) = (ye)((xa)((xa2)y))

= (xa)((ye)((xa2)y)) = (xa)(((ye)(x(aa)))y) = (xa)(((ye)(a(xa)))y)

= (xa)((a((ye)(xa)))y) = (xa)((y((ye)(xa)))a) ≤ (xa)p,


where p ≤ ((y((ye)(xa)))a) (by reflexive property) and therefore, we have

(S µ)2(a) =∨

a≤(xa)p

(S µ)(xa) ∧ (S µ)(p)

≥ (S µ)(xa) ∧ (S µ)(p)=

∨xa≤xa

S(x) ∧ µ(a) ∧∨

p≤(y(y(xa)))a

S(y((ye)(xa))) ∧ µ(a)

≥ S(x) ∧ µ(a) ∧ S(y((ye)(xa))) ∧ µ(a) = µ(a).

Thus we get the required µ = (S µ)2.(ii) =⇒ (i) : Let µ = (S µ)2 holds for any i-v fuzzy two-sided ideal µ of S, then by

using Lemma 4.9 and given assumption, we have

µ = (S µ)2 ⊆ (µ)2 = µ µ ⊆ S µ ⊆ µ.

Which shows that µ = µµ, therefore by using Lemma 4.13, S is an intra-regular.

Lemma 4.15. Let S be an intra-regular ordered LA-semigroup with left identity andlet µ and γ be any i-v fuzzy two-sided ideals of S, then µ γ = µ ∩ γ.

Proof. Assume that µ and γ are any i-v fuzzy two-sided ideals of an intra-regular orderedLA-semigroup S with left identity, then by using Lemma 4.9, we have µ γ ⊆ µ ∩ γ.Let a ∈ S, then there exist x, y ∈ S such that a ≤ (xa2)y. Now by using (4) and (2), wehave

a ≤ (x(aa))y = (a(xa))(ey) = (ae)((xa)y).

Therefore, we have

(µ γ)(a) =∨

a≤(ae)((xa)y)

µ(ae) ∧ γ((xa)y) ≥ µ(ae) ∧ γ((xa)y)

≥ µ(a) ∧ γ(a) = (µ ∩ γ)(a).

which gives us µ γ ⊇ µ ∩ γ and therefore µ γ = µ ∩ γ.

Lemma 4.16. In an intra-regular ordered LA-semigroup S, S µ = µ and µ S = µholds for an i-v fuzzy subset µ of S.

Proof. Assume that µ is an i-v fuzzy subset of an intra-regular ordered LA-semigroupS and let a ∈ S, then there exist x, y ∈ S such that a ≤ (xa2)y. Now by using (4) and(1), we have

a ≤ (x(aa))y = (a(xa))y = (y(xa))a.

Therefore

(S µ)(a) =∨

a≤(y(xa))a

S(y(xa)) ∧ µ(a) =∨

a≤(y(xa))a

1 ∧ µ(a)

=∨

a≤(y(xa))a

µ(a) = µ(a),


which shows that S µ = µ.Now by using (3) and (4), we have

a ≤ (xa2)(ey) = (ye)(a2x) = (aa)((ye)x) = (x(ye))(aa) = a((x(ye))a).

Therefore

(µ S)(a) =∨

a≤a((x(ye))a)

µ(a) ∧ S((x(ye))a) =∨

a≤a((x(ye))a)

µ(a) ∧ 1

=∨

a≤a((x(ye))a)

µ(a) = µ(a),

which shows that µ S = µ.

Corollary 4.17. In an intra-regular ordered LA-semigroup S, S µ = µ and µ S = µholds for every i-v fuzzy two-sided ideal of S.

Theorem 4.18. The set of i-v fuzzy two-sided ideals of an intra-regular ordered LA-semigroup S with left identity forms a semilattice structure with identity S.

Proof. Assume that I is the set of all i-v fuzzy two-sided ideals of an intra-regular orderedLA-semigroup S and let µ, γ and α be in I. Clearly I is closed and by Theorem 4.13,µ2 = µ. Now by using Lemma 4.15, we get µ γ = γ µ and therefore, we have

(µ γ) α = (γ µ) α = (α µ) γ = (µ α) γ = (γ α) µ = µ (γ α).

It is easy to see from Corollary 4.17 that S is an identity in I.

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ordered $\\mathcal{la}$-semigroups in terms of interval valued fuzzy ideals

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