volume 113 no. 2 2017, 299-325 - ijpam · ucts, fuzzy relations, and fuzzy functions. in addition,...

International Journal of Pure and Applied Mathematics

Volume 113 No. 2 2017, 299-325ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: 10.12732/ijpam.v113i2.10

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STRUCTURES OF GENERALIZED FUZZY SETS

IN NON-ASSOCIATIVE RINGS

Inayatur Rehman1 §, Muhammad Gulistan2,Muhammad Asif Gondal3, Shah Nawaz4

1,3Department of Mathematics and SciencesCollege of Arts and Applied Sciences

Dhofar UniversitySalalah, OMAN

2,4Department of MathematicsHazara University

Mansehra, PAKISTAN

Abstract: Since the introduction of the concept of fuzzy sets, the theoretical application

of fuzzy sets has been restricted to associative algebraic structures (groups, semigroups, as-

sociative rings, semi-rings etc). In addition, the study of fuzzy sets, where the base set is a

commutative structure, has attracted the attention of many researchers. On the other hand

there are many sets which are naturally endowed with two compatible binary operations form-

ing a non-associative ring and we may dig out examples which investigate a non-associative

structure in the context of fuzzy sets. Intuitively one can apply the concept of fuzzy sets

to non-commutative and non-associative structures.In this paper, we introduce the concept

of (α, β)-fuzzy ideals in LA-rings (a non-associative structure). We discuss the important

features of a non-associative regular LA-ring by using (α, β) -fuzzy bi-ideals, (α, β)-fuzzy gen-

eralized bi-ideals, (α, β)-fuzzy quasi-ideals, and (α, β)-fuzzy interior ideals. Ultimately, we

identify upper and lower parts of these structures and characterize regular LA-rings using the

identified properties of these structures.

AMS Subject Classification: 17D99

Key Words: LA-rings, Fuzzy LA-rings, (α, β)-fuzzy ideals, Regular LA-Rings

Received: November 13, 2016

Revised: January 20, 2017

Published: March 19, 2017

c© 2017 Academic Publications, Ltd.

url: www.acadpubl.eu

§Correspondence author

300 I. Rehman at all

1. Introduction

After fuzzy sets were introduced by Zadeh [24], the concepts of classical math-ematics began to be reconsidered. The theory of fuzzy semigroups was firstdeveloped by Kuroki [13]. Mordeson et al. (2002) deal with the application ofthe fuzzy approach to the concept of automata and formal languages. Later,in 2003, Mordeson et al. established theoretical results for fuzzy semigroups,and used these concepts in fuzzy coding, fuzzy finite-state machines, and fuzzylanguages. In 2004, Murali [14] defined fuzzy points belonging to fuzzy subsets.

Bhakat et al. [2] introduced the concept of (∈,∈ ∨q)-fuzzy subgroups, andalso discussed (α, β)-fuzzy subgroups. The concept of (∈,∈ ∨qk)-fuzzy sub-groups is a valuable generalization of Rosenfeld’s fuzzy subgroups provided byJun [10]. In 2006, Jun et al. [11] published a study on the (α, β)-fuzzy interiorideals of a semigroup. Davvaz (2006) investigated the concept of (∈,∈ ∨qk)-fuzzy sub-near rings and the ideal characteristics of a near ring. Kazanci et al.[12] studied (∈,∈ ∨qk)-fuzzy bi-ideal of semigroup. Shabir et al. [16] charac-terised semigroups based on (∈,∈ ∨qk)-fuzzy ideals. Yin et al. [23] introducedmore general forms of .(∈,∈ ∨q)-fuzzy filters and defined (∈γ ,∈γ ∨qδ)-fuzzyfilters, providing some interesting results relating to these notions. Gulistan etal. [?] developed the concept of the Hv-LA-semigroup, and in an another paper(2015) they discussed (α, β)-fuzzy KU-ideals of KU-algebras. A generalisationof fuzzy sets was provided by Atanassov [1] (1986), using the concept of intu-itionistic fuzzy sets. Fuzzy rings and fuzzy ideals were discussed by Dib et al.[4]. Earlier, Dib et al. [5] examined the properties of fuzzy Cartesian prod-ucts, fuzzy relations, and fuzzy functions. In addition, Volf [22] investigatedthe properties of fuzzy subfields.

Left Almost ring (LA-ring) is actually an off shoot of LA-semigroup andLA-group. It is a non-commutative and non-associative structure. Due toits peculiar characteristics, it has been emerging as useful non-associative classwhich intuitively would have reasonable contribution to enhance non-associativering theory. By an LA-ring, we mean a non-empty set R with at least twoelements such that (R,+) is an LA-group, (R, ·) is an LA-semigroup, both leftand right distributive laws hold. The Left Almost ring (LA-ring) of finitely non-zero functions was introduced by Shah et al. (2010). Later, Shah et al. [17]investigated some characteristics of LA-rings based on their ideal. The conceptof ideal theory provided a platform to investigate the application of fuzzy sets,intuitionistic fuzzy sets, and soft sets in LA-rings. Moreover, Shah et al. [18]defined intuitionistic fuzzy sets in LA-rings, and provided some useful results.Rehman et al. [15] did some computational work using Mace4, exploring some

STRUCTURES OF GENERALIZED FUZZY SETS... 301

interesting characteristics of LA-rings. Shah et al.[19] discussed soft M-systemsin a class of non-associative rings. Recently, Shah et al. [20] have applied theconcept of soft sets to LA-rings, and have established some useful results fornon-associative soft-set theory.

In this paper, we introduce the concept of (α, β)-fuzzy ideals in LA-rings.We discuss the important features of a non-associative regular LA-ring using(α, β)-fuzzy bi-ideals and (α, β)- fuzzy generalised bi-ideals.

2. Generalized Fuzzy Ideals

Let R be an LA-ring. A fuzzy subset f over R is a function from R into unitclosed interval [0, 1], i.e. f : R −→ [0, 1]. A fuzzy subset f over R of the form

f (y) =

{

t ∈ (0, 1] if y = x0 if y 6= x

, is said to be a fuzzy point with support x

and value t and is denoted by xt. For a fuzzy point xt and fuzzy set f over Rwe have

(i) xt ∈ f if f(x) ≥ t.

(ii) xtqf if f(x) + t > 1.

(iii) xt ∈ ∨qf if xt ∈ f or xtqf.

(iv) xt ∈ ∧qf if xt ∈ f and xtqf.

(v) xtqkf if f(x) + t+ k > 1, where t, k ∈ (0, 1].

(vi) xt ∈γ f if f(x) ≥ t > γ and xtqδf if f(x) + t > 2δ, where t ∈ (γ, 1] andγ, δ ∈ (0, 1].

Definition 1. An (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy subsetf of an LA-ring R is called an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzyLA-subring of R, if xt1 ∈ f, yt2 ∈ f (resp., xt1 ∈ f, yt2 ∈ f and xt1 ∈γ f,and yt2 ∈γ f) ⇒ (x− y)min{t1,t2}

∈ ∨qf(resp., (x− y)min{t1,t2}∈ ∨qkf and

(x− y)min{t1,t2}∈γ ∨qδf) and (xy)min{t1,t2}

∈ ∨qf(resp., (xy)min{t1,t2}∈ ∨qkf

and (xy)min{t1,t2}∈γ ∨qδf) where we required k ∈ (0, 1] for (∈,∈ ∨qk)-fuzzy

subset and t1, t2,∈ (γ, 1] for (∈γ ,∈γ ∨qδ)-fuzzy subset.

Corollary 2. Every (∈,∈ ∨q)-fuzzy LA-subring is an (∈,∈ ∨qk)-fuzzyLA-subring and every (∈,∈ ∨qk)-fuzzy LA-subring is an (∈γ ,∈γ ∨qδ)-fuzzyLA-subring of R, but converse is not true.


Example 1. R = {0, 1, 2, 3, 4, 5, 6, 7} is an LA-ring defined by the follow-ing Cayley table.

+ 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 71 2 0 3 1 6 4 7 52 1 3 0 2 5 7 4 63 3 2 1 0 7 6 5 44 4 5 6 7 0 1 2 35 6 4 7 5 2 0 3 16 5 7 4 6 1 3 0 27 7 6 5 4 3 2 1 0

· 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 01 0 4 4 0 0 4 4 02 0 4 4 0 0 4 4 03 0 0 0 0 0 0 0 04 0 3 3 0 0 3 3 05 0 7 7 0 0 7 7 06 0 7 7 0 0 7 7 07 0 3 3 0 0 3 3 0

Let us define f (0) = 0.81, f (1) = 0.8, f (2) = 0.75, f (3) = 0.7, f (4) = 0.65,f (5) = 0.6, f (6) = 0.55, f (7) = 0.55 and let t1 = 0.54 and t2 = 0.53. Thenby routine calculation it can be easily seen that f is an (∈,∈ ∨q)−fuzzy LA-subring of R, it is an (∈ ∨qk)-fuzzy LA-subring and it is also an (∈γ ∨qδ)-fuzzyLA-subring of R, where k,γ, δ ∈ [0, 1] with condition that γ < δ. If we definethe fuzzy sets as

f (0) = 0.8, f (1) = 0.8, f (2) = 0.7, f (3) = 0.7,

f (4) = 0.6, f (5) = 0.6, f (6) = 0.5, f (7) = 0.49

and let t1 = 0.25 and t2 = 0.31 with γ = 0.2 and δ = 0.25 then it is obviousthat

1) f is an (∈0.2,∈0.2 ∨q0.25)-fuzzy LA-subring of R.

2) f is not an (∈,∈ ∨q0.25)-fuzzy LA-subring of R because 00.25 ∈ f and70.31 ∈ f but (0− 7)0.25∧0.31 = 70.25∈ ∨q0.25f.

3) f is not an (∈,∈ ∨q)-fuzzy LA-subring of R because 00.25 ∈ f and 70.31 ∈ fbut (0− 7)0.25∧0.31 = 70.25∈ ∨qf.

Definition 3. An (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy subsetf of an LA-ring R is called an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzyleft (resp: right) ideal of R, if it satisfies, yt ∈ f(resp., yt ∈ f, yt ∈γ f) andx ∈ R =⇒ (xy)t ∈ ∨qf(resp., (xy)t ∈ ∨qkf , (xy)t ∈γ ∨qδf ), for all x, y ∈ Rwith t ∈ [γ, 1) for (∈γ ,∈γ ∨qδ))-fuzzy set where we required k, δ ∈ (0, 1] for(∈,∈ ∨qk)-fuzzy subset.

The concept of (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy right idealsis defined in the similar way.


Definition 4. A fuzzy subset f of an LA-ringR is called an (∈,∈ ∨q) (resp.,(∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy ideal of R if it is both an (∈,∈ ∨q) (resp.,(∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy left ideal and (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ

,∈γ ∨qδ))-fuzzy right ideal of R.

Corollary 5. Every (∈,∈ ∨q)-fuzzy ideal is an (∈,∈ ∨qk)-fuzzy ideal andevery (∈,∈ ∨qk)-fuzzy ideal is an (∈γ ,∈γ ∨qδ)-fuzzy ideal of R, but the converseis not true.

Definition 6. An (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy LA-subring f of an LA-ring R is called an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ

∨qδ))-fuzzy bi ideal of R, if for x, y, z ∈ R and for all t3,t4, k ∈ (0, 1] , xt3 ∈f, zt4 ∈ f(resp., xt3 ∈ f, zt4 ∈ f, xt3 ∈γ f, zt4 ∈γ f) =⇒ ((xy) z)min{t3,t4}

∈∨qf(resp., ((xy) z)min{t3,t4}

∈ ∨qkf, ((xy) z)min{t3,t4}∈γ ∨qδf) with t3,t4 ∈

[γ, 1), δ ∈ (0, 1] for (∈γ ,∈γ ∨qδ))-fuzzy set and k ∈ (0, 1] for (∈,∈ ∨qk)-fuzzysubset.

Corollary 7. Every (∈,∈ ∨q)-fuzzy bi-ideal is an (∈,∈ ∨qk)-fuzzy bi-idealand every (∈,∈ ∨qk)-fuzzy bi-ideal is an (∈γ ,∈γ ∨qδ)-fuzzy bi-ideal of R, butthe converse is not true.

Definition 8. An (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy set fof an LA-ring R is called an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzygeneralized bi-ideal of R, if for x, y, z ∈ R and for all t3,t4, k ∈ (0, 1] , xt3 ∈f, zt4 ∈ f(resp., xt3 ∈ f, zt4 ∈ f, xt3 ∈γ f, zt4 ∈γ f) =⇒ ((xy) z)min{t3,t4}

∈∨qf(resp., ((xy) z)min{t3,t4}

∈ ∨qkf, ((xy) z)min{t3,t4}∈γ ∨qδf) with t3,t4 ∈

[γ, 1), δ ∈ (0, 1] for (∈γ ,∈γ ∨qδ))-fuzzy set and k ∈ (0, 1] for (∈,∈ ∨qk)-fuzzysubset.

Corollary 9. Every (∈,∈ ∨q)-fuzzy generalized bi-ideal is an (∈,∈ ∨qk)-fuzzy generalized bi-ideal and every (∈,∈ ∨qk)-fuzzy generalized bi-ideal is an(∈γ ,∈γ ∨qδ)-fuzzy generalized bi-ideal of R, but the converse is not true.

Definition 10. An (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy LA-subring f of an LA-ring R is called an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ

∨qδ))-fuzzy interior ideal of R, if for x, a, y ∈ R and for all t, k ∈ (0, 1] , at ∈f, (resp., at ∈ f, at ∈γ f) =⇒ ((xa) y)t ∈ ∨qf(resp., ((xa) y)t ∈ ∨qkf, ((xa) y)t ∈γ

∨qδf) with t ∈ [γ, 1), δ ∈ (0, 1] for (∈γ ,∈γ ∨qδ))-fuzzy set and k ∈ (0, 1] for(∈,∈ ∨qk)-fuzzy subset.

Corollary 11. Every (∈,∈ ∨q)-fuzzy interior ideal is an (∈,∈ ∨qk)-fuzzyinterior ideal and every (∈,∈ ∨qk)-fuzzy interior ideal is an (∈γ ,∈γ ∨qδ)-fuzzyinterior ideal of R, but the converse is not true.


Theorem 12. If f is a nonzero (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy LA-subring (resp.,fuzzy ideal, fuzzy bi-ideal, fuzzy generalized bi-ideal,fuzzy interior ideal ) of R, then the set f0 = {x ∈ R | f (x) > 0} is an LA-subring(resp., ideal, bi-ideal, generalized bi-ideal, interior ideal ) of R.

Proof. If f is a nonzero (∈,∈ ∨q)-fuzzy LA-subring and x, y ∈ f0, thenf (x) > 0 and f (y) > 0. Let f (xy) = 0 and f (x− y) = 0. Then xf(x) ∈f and yf(y) ∈ f but f (xy) = 0 < min {f (x) , f (y)} , f (x− y) = 0 <min {f (x) , f (y)} and f (xy) + min {f (x) , f (y)} ≤ 0 + 1 = 1, f (x− y) +min {f (x) , f (y)} ≤ 0 + 1 = 1. So

(xy)min{f(x),f(y)} ∈ ∨qf

and(x− y)min{f(x),f(y)} ∈ ∨qf

a contradiction. Hence f (xy) > 0, f (x− y) > 0, that is xy ∈ f0 and x−y ∈ f0.Also x1qf and y1qf but (xy)1 ∈ ∨qf and (x− y)1 ∈ ∨qf. Hence f (xy) > 0,f (x− y) > 0, that is, xy ∈ f0 and x − y ∈ f0. Thus f0 is an LA-subring ofR. Similarly the case for (∈,∈ ∨qk) and (∈γ ,∈γ ∨qδ)-fuzzy sets can easily beseen.

Theorem 13. If f is a fuzzy subset of an LA-ring R, then 1) f is an(∈,∈ ∨q)-fuzzy LA-subring of R if and only if f (xy) ≥ min {f (x) , f (y) , 0.5}and f (x− y) ≥ min {f (x) , f (y) , 0.5}. 2) f is an (∈,∈ ∨qk)-fuzzy LA-subringof R if and only if f (xy) ≥ min

{

f (x) , f (y) , 1−k2

}

and

f (x− y) ≥ min

{

f (x) , f (y) ,1− k

2

}

. 3) f is an (∈γ ,∈γ ∨qδ)-fuzzy LA-subring of R if and only if max{f(xy), γ} ≥min{f(x), f(y), δ} and max{f(x− y), γ} ≥ min{f(x), f(y), δ}.

Proof. 1) Let f be an (∈,∈ ∨q)-fuzzy LA-subring of R. Assume thatthere exist x, y ∈ R such that f(xy) < min{f(x), f(y), 0.5} and f(x − y) <min{f(x), f(y), 0.5}. Choose t ∈ (0, 1] such that f(xy) < t ≤ min{f(x), f(y), 0.5}and f(x− y) < t ≤ min{f(x), f(y), 0.5}.Then xt ∈ f and yt ∈ f but f(xy) < twith f(x−y) < t and f(xy)+t+0.5 = 1 with f(x−y)+t+0.5 = 1, so (xy)t∈ ∨qfand (x−y)t∈ ∨qf, which is a contradiction. Hence f(xy) ≥ min{f(x), f(y), 0.5}and f (x− y) ≥ min {f (x) , f (y) , 0.5}. Conversely, assume that f (xy) ≥min {f (x) , f (y) , 0.5} and f (x− y) ≥ min {f (x) , f (y) , 0.5}. Let xt ∈ fand yr ∈ f for t, r ∈ (0, 1]. Then f(x) ≥ t and f(y) ≥ r. Now f(xy) ≥


min{f(x), f(y), 0.5} ≥ min{t, r, 0.5} and f(x − y) ≥ min{f(x), f(y), 0.5} ≥min{t, r, 0.5}. We consider two cases: Case (i): If t ∧ r > 0.5, then f(xy) +t ∧ r + 0.5 > 0.5 + 0.5 = 1 and f(x − y) + t ∧ r + 0.5 > 0.5 + 0.5 = 1, whichimplies that (xy)min{t,r}qf and (x− y)min{t,r}qf. Case (ii): If t∧ r ≤ 1−k

2 , thenf(xy) ≥ t ∧ r and f(x− y) ≥ t ∧ r. So (xy)min{t,r} ∈ f and (x− y)min{t,r} ∈ f.Thus in both cases (xy)min{t,r} ∈ ∨qf and (x− y)min{t,r} ∈ ∨qf. Therefore f isan (∈,∈ ∨qk)-fuzzy LA-subring of R. 2) The proof is similar to the proof of 1).3) Let f be (∈γ ,∈γ ∨qδ)-fuzzy LA-subring of R. Assume that there exist x, y∈ R and t ∈ (γ, 1], such that

max{f(xy), γ} < t ≤ min{f(x), f(y), δ}

and

max{f(x− y), γ} < t ≤ min{f(x), f(y), δ}.

Thus max{f(xy), γ} < t and max{f(x − y), γ} < t, this implies that f(xy) <t ≤ γ and f(x−y) < t ≤ γ, which further implies that (xy)min{t,s}∈γ ∨qδf with(x−y)min{t,s}∈γ ∨qδf . As min{f(x), f(y), δ} ≥ t, therefore min{f(x), f(y)} ≥ tthis implies that f(x) ≥ t > γ, f(y) ≥ t > γ, implies that xt ∈γ f , ys ∈γ

f but (xy)min{t,s}∈γ ∨qδf and (x − y)min{t,s}∈γ ∨qδf a contradiction to thedefinition. Hence max{f(xy), γ} ≥ min{f(x), f(y), δ} and max{f(x− y), γ} ≥min{f(x), f(y), δ} for all x, y ∈ R. Conversely, assume that there exist x, y ∈R and t, s ∈ (γ, 1] such that xt ∈γ f , ys ∈γ f. By definition we write f(x)≥ t > γ, f(y) ≥ s > γ, so max{f(xy), δ} ≥ min{f(x), f(y), δ} and max{f(x−y), δ} ≥ min{f(x), f(y), δ} this implies that f(xy) ≥ min{t, s, δ} and f(x−y) ≥min{t, s, δ}. Here we have two cases. Case (i): If {t, s} ≤ δ, then f(xy) ≥min{t, s} > γ and f(x− y) ≥ min{t, s} > γ. This implies that (xy)min{t,s} ∈γ fand (x−y)min{t,s} ∈γ f. Case (ii): If {t, s} > δ, then f(xy)+min{t, s} > 2δ andf(x−y)+min{t, s} > 2δ. This implies that (xy)min{t,s}qδf and (x−y)min{t,s}qδf.From both cases we get (xy)min{t,s} ∈γ ∨qδf and (x− y)min{t,s} ∈γ ∨qδf for allx, y in S. Hence f is an (∈γ ,∈γ ∨qδ)-fuzzy LA-subring of R.

Theorem 14. If f is a fuzzy subset of R, then 1) f is an (∈,∈ ∨q)-fuzzy ideal of R if and only if f (xy) ≥ min {f (x) , f (y) , 0.5} and f (x− y) ≥min {f (x) , f (y) , 0.5}. 2) f is an (∈,∈ ∨qk)-fuzzy ideal of R if and only iff (xy) ≥ min

{

f (x) , f (y) , 1−k2

}

and f (x− y) ≥ min{

f (x) , f (y) , 1−k2

}

. 3) fis an (∈γ ,∈γ ∨qδ)-fuzzy ideal of S if and only if

max{f(xy), γ} ≥ min{f(x), f(y), δ}, max{f(x− y), γ} ≥ min{f(x), f(y), δ},

where γ, δ ∈ [0, 1].


Proof. The same as the proof of the Theorem 13.

Theorem 15. If f is a fuzzy subset of R, then 1) f is an (∈,∈ ∨q)-fuzzy generalized bi-ideal of R if and only if f ((xy) z) ≥ min {f (x) , f (z) , 0.5}for all x, y, z ∈ R. 2) f is an (∈,∈ ∨qk)-fuzzy generalized bi-ideal of R ifand only if f ((xy) z) ≥ min

{

f (x) , f (z) , 1−k2

}

for all x, y, z ∈ R. 3) f is an(∈γ ,∈γ ∨qδ)-fuzzy generalized bi-ideal of R if and only if max{f ((xy) z) , γ} ≥min {f (x) , f (z) , δ} for all x, y, z ∈ R.

Proof. The proof is straightforward.

Theorem 16. If f is a fuzzy LA-subring of R, then 1) f is an (∈,∈ ∨q)-fuzzy bi-ideal of R if and only if f ((xy) z) ≥ min {f (x) , f (z) , 0.5} for allx, y, z ∈ R. 2) f is an (∈,∈ ∨qk)-fuzzy bi-ideal of R if and only if f ((xy) z) ≥min

{

f (x) , f (z) , 1−k2

}

for all x, y, z ∈ R. 3) f is an (∈γ ,∈γ ∨qδ)-fuzzy bi-idealof R if and only if max{f ((xy) z) , γ} ≥ min {f (x) , f (z) , δ} for all x, y, z ∈ R.


Theorem 17. If f is a fuzzy LA-subring of an LA-ring R, then 1) f isan (∈,∈ ∨q)-fuzzy interior-ideal of R if and only if f ((xy) z) ≥ min {f (y) , 0.5}for all x, y, z ∈ R. 2) f is an (∈,∈ ∨qk)-fuzzy interior-ideal of R if and only iff ((xy) z) ≥ min

{

f (y) , 1−k2

}

for all x, y, z ∈ R. 3) f is an (∈γ ,∈γ ∨qδ)-fuzzyinterior-ideal of R if and only if max{f ((xy) z) , γ} ≥ min {f (y) , δ} for allx, y, z ∈ R.


Definition 18. If f is a fuzzy subset of an LA-ring R, then 1) f is calledan (∈,∈ ∨q)-fuzzy quasi-ideal of R, if it satisfies,

f (x) ≥ min {((f ◦ δ) , (δ ◦ f)) (x) , 0.5} ,

where δ is the fuzzy subset of R mapping every element of R on 1. 2) f iscalled an (∈,∈ ∨qk)-fuzzy quasi-ideal of R, if it satisfies,

f (x) ≥ min

{

((f ◦ δ) , (δ ◦ f)) (x) ,1− k

2

}

,

where δ is the fuzzy subset of R mapping every element of R on 1. 3) f iscalled an (∈γ ,∈γ ∨qδ)-fuzzy quasi-ideal of R, if it satisfies, max{f (x) , γ} ≥min {((f ◦ δ) , (δ ◦ f)) (x) , δ}, where δ is the fuzzy subset of R mapping everyelement of R on 1.


Theorem 19. Intersection of (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy (LA-subrings, ideals) of an LA-ring R is an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) ,(∈γ ,∈γ ∨qδ))-fuzzy(LA-subring, ideal) of R.

Proof. Let {fi}i∈I be a family of (∈,∈ ∨q)-fuzzy ideals of R. Let x, y ∈ R.Then

(∧i∈If) (xy) = ∧i∈I (fi (xy) , ) and (∧i∈If) (x− y) = ∧i∈I (fi (x− y)) .

Since each fi is an (∈,∈ ∨q)-fuzzy ideal of R, so

fi (xy) ≥ min {fi (x) , fi (y) , 0.5}

andfi (x− y) ≥ min {fi (x) , fi (y) , 0.5} for all i ∈ I,

thus

(∧i∈I) f (xy) = ∧i∈I (fi (xy)) ≥ ∧i∈I (fi (x) , fi (y) ∧ 0.5)

= (∧i∈Ifi (x) ,∧i∈Ifi (y)) ∧ 0.5 = (∧i∈Ifi) (x ∧ y) ∧ 0.5

and

(∧i∈I) f (x− y) = ∧i∈I (fi (x− y)) ≥ ∧i∈I (fi (x) ∧ fi (y) ∧ 0.5)

= (∧i∈Ifi (x) ∧ fi (y)) ∧ 0.5 = (∧i∈Ifi) (x ∧ y) ∧ 0.5.

Hence ∧i∈If is an (∈,∈ ∨q)-fuzzy ideals of an LA-ring R. Similarly it caneasily be proved for other type of fuzzy ideals.

Theorem 20. Every (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzygeneralized bi-ideal of a regular LA-ring R is an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) ,(∈γ ,∈γ ∨qδ))-fuzzy bi-ideal of R.

Proof. Let f be any (∈,∈ ∨q)- fuzzy generalized bi-ideal of R and let a, bbe any elements of R. Then there exists an element x ∈ R such that b = (bx) b.Thus we have f (ab) = f (a (bx) b) ≥ min {f (a) , f (b) , 0.5} . This shows that fis an (∈,∈ ∨q)-fuzzy sub LA-ring of R and so f is an (∈,∈ ∨q)-fuzzy bi-idealof R.

Theorem 21. Let f be an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy quasi-ideal of an LA-ring R. Then the set f0 = {x ∈ R | f (x) > 0} is aquasi-ideal of R.


Proof. In order to show that f0 is a quasi-ideal of R, we have to show thatRf0 ∩ f0R ⊆ f0. Let a ∈ Rf0 ∩ f0R. This implies that a ∈ Rf0 and a ∈ f0R.So a = rx and a = yt for some r, t ∈ R and x, y ∈ f0. Thus f (x) > 0 andf (y) > 0. Now f (a) ≥ min {(f ◦ δ) (a) , (δ ◦ f) (a) , 0.5} . Since

(δ ◦ f) (a) =∨

a=

n∑

i=1

piqi

{

n∧

i=1

{δ (pi) ∧ f (qi)}

}

≥ {δ (s) ∧ f (x)} = f (x) .

Similarly (f ◦ δ) (a) ≥ f (y) . Thus

f (a) ≥ min {(f ◦ δ) (a) , (δ ◦ f) (a) , 0.5} ≥ min {f (x) , f (y) , 0.5} > 0.

This implies that a ∈ f0. Hence f0 is a quasi-ideal of R.

Corollary 22. Every fuzzy quasi-ideal ofR is an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) ,(∈γ ,∈γ ∨qδ))-fuzzy quasi ideal of R.

Theorem 23. Every (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy leftideal of R is an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy quasi-ideal ofR where we required k ∈ (0, 1] for (∈,∈ ∨qk)-fuzzy subset and t1, t2,∈ (γ, 1] for(∈γ ,∈γ ∨qδ)-fuzzy subset.

Proof. Let x ∈ R, then

(δ ◦ f) (x) =∨

x=

n∑

i=1

yizi

{

n∧

i=1

{R (yi) ∧ f (zi)}

}

=∨

x=

n∑

i=1

yizi

f (zi) .

This implies that

(δ ◦ f) (x)∧0.5 =

∨

x=

n∑

i=1

yizi

f (zi)

∧0.5 =∨

x=

n∑

i=1

yizi

(f (zi) ∧ 0.5) ≤ f (yz) = f (x) .

Thus (δ ◦ f) (x) ∧ 0.5 ≤ f (x) . Hence

f (x) ≥ (δ ◦ f) (x) ∧ 0.5 ≥ min {(f ◦ δ) (x) , (δ ◦ f) (x) , 0.5} .


Thus f is an (∈,∈ ∨q)-fuzzy quasi-ideal of R. Similarly we can show thatevery (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy right ideal of R is an(∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy quasi-ideal of R.

Theorem 24. If R is an LA-ring with left identity e such that (xe)R =xR, then every (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy quasi-ideal ofR is an (∈,∈ ∨q) (resp., (∈,∈ ∨qk) , (∈γ ,∈γ ∨qδ))-fuzzy bi-ideal of R.

Proof. Suppose f is an (∈,∈ ∨q)-fuzzy quasi-ideal of R. Now

f (xy) ≥ (f ◦ δ) (xy) ∧ (δ ◦ f) (xy) ∧ 0.5

=

∨

xy=

n∑

i=1

aibi

{

n∧

i=1

{f (ai) ∧ δ (bi)}

}

∧

∨

xy=

n∑

i=1

piqi

{

n∧

i=1

{δ (pi) ∧ f (qi)}

}

∧ 0.5

≥ [f (x) ∧ δ (y)] ∧ [δ (x) ∧ f (y)] ∧ 0.5

≥ [f (x) ∧ 1] ∧ [1 ∧ f (y)] ∧ 0.5 = f (x) ∧ f (y) ∧ 0.5.

So

f (xy) ≥ min {f (x) , f (y) , 0.5} .

Also

f ((xy) z) ≥ (f ◦ δ) ((xy) z) ∧ (δ ◦ f) ((xy) z) ∧ 0.5.

Now,

(δ ◦ f) ((xy) z) =∨

(xy)z=

n∑

i=1

piqi

{

n∧

i=1

δ (pi) ∧ f (qi)

}


=∨

(xy)z=

n∑

i=1

piqi

{

n∧

i=1

{1 ∧ f (qi)}

}

≥ f (z) .

Then(δ ◦ f) ((xy) z) ≥ f (z) .

Also

(f ◦ δ) ((xy) z) =∨

(xy)z=

n∑

i=1

aibi

{

n∧

i=1

{f (ai) ∧ δ (bi)}

}

=∨

(xy)z=

n∑

i=1

aibi

{

n∧

i=1

{f (ai) ∧ 1}

}

.

Now,(xy) z = (xy) (ez) = (xe) (yz) ∈ (xe)R = xR.

So (xy) z = xt for some t ∈ R. Then

(f ◦ δ) ((xy) z) =∨

xt=

∑

aibi

{

n∧

i=1

{f (a) ∧ 1}

}

≥ f (x) .

Thusf ((xy) z) ≥ f (x) ∧ f (z) ∧ 0.5.

Hence f is an (∈,∈ ∨q)-fuzzy bi-ideal of R.

3. Lower and Upper Parts of (∈,∈ ∨q)-Fuzzy Ideals

In this section we identify lower and upper parts of (∈,∈ ∨q)-fuzzy ideals ofLA-rings and explore some interesting properties of LA-rings. At the end ofthis section, we characterize regular LA-rings by using the properties of

(∈,∈ ∨q)-fuzzy ideals and lower and upper parts of (∈,∈ ∨q)-fuzzy ideals.


Definition 25. Let f be a fuzzy subset of an LA-ring R. We define theupper part f+ and the lower part f− of f as follows, f+ (x) = f (x) ∨ 0.5 andf− (x) = f (x) ∧ 0.5.

Theorem 26. If f and g are fuzzy subsets of an LA-ring R, then

(1) (f ∧ g)− = (f− ∧ g−)

(2) (f ∨ g)− = (f− ∨ g−)

(3) (f ◦ g)− = (f− ◦ g−)

Proof. (1) For all a ∈ R,

(f ∧ g)− (a) = (f ∧ g) (a) ∧ 0.5 = f (a) ∧ g (a) ∧ 0.5 = (f (a) ∧ 0.5) ∧ (g (a) ∧ 0.5)

= f− (a) ∧ g− (a) =(

f− ∧ g−)

(a) .

(2) For all a ∈ R,

(f ∨ g)− (a) = (f ∨ g) (a) ∧ 0.5 = (f (a) ∨ g (a)) ∧ 0.5 = (f (a) ∧ 0.5) ∨ (g (a) ∧ 0.5)

= f− (a) ∨ g− (a) =(

f− ∨ g−)

(a) .

(3) If a is not expressible as a = bc for some b, c ∈ R,

then

(f ◦ g) (a) =∨

a=

n∑

i=1

bici

{

n∧

i=1

{f (bi) ∧ g (ci)}

}

= 0 ∧ 0 = 0.

Thus (f ◦ g)− (a) = (f ◦ g) (a) ∧ 0.5 = 0. Since a is not expressible as a = bc,so,

(

f− ◦ g−)

(a) =∨

a=

n∑

i=1

bici

{

n∧

i=1

{

f− (bi) ∧ g − (ci)}

}

=∨

a=

n∑

i=1

bici

{

n∧

i=1

{f (bi) ∧ g (ci) ∧ 0.5}

}

= 0.


Thus in this case (f ◦ g)− = (f− ◦ g−) . if a is expressible as a = xy for somex, y ∈ R, then

(f ◦ g)− (a) = (f ◦ g) (a) ∧ 0.5 =∨

a=

n∑

i=1

xiyi

{

n∧

i=1

{f (xi) ∧ g (yi)}

}

∧ 0.5

=∨

a=

n∑

i=1

xiyi

{

n∧

i=1

{(f (xi) ∧ 0.5) ∧ (g (yi) ∧ 0.5)}

}

=∨

a=

n∑

i=1

xiyi

{

f− (xi) ∧ g− (yi)}

=(

f− ◦ g−)

(a) .

Theorem 27. If f and g are fuzzy subsets of an LA-ring R, then thefollowing properties hold.

(1) (f ∧ g)+ = (f+ ∧ g+)(2) (f ∨ g)+ = (f+ ∨ g+)(3) (f ◦ g)+ ≥ (f+ ◦ g+) .If every element x of R is expressible as x = bc, then (f ◦ g)+ = (f+ ◦ g+) .

Proof. (1) For all a ∈ R,

(f ∧ g)+ (a) = (f ∧ g) (a) ∨ 0.5 = (f (a) ∧ g (a)) ∨ 0.5 = (f (a) ∨ 0.5) ∧ (g (a) ∨ 0.5)

= f+ (a) ∧ g+ (a) =(

f+ ∧ g+)

(a) .

(2) For all a ∈ R,

(f ∨ g)+ (a) = (f ∨ g) (a) ∨ 0.5 = f (a) ∨ g (a) ∨ 0.5 = (f (a) ∨ 0.5) ∨ (g (a) ∨ 0.5)

= f+ (a) ∨ g+ (a) =(

f+ ∨ g+)

(a) .

(3) If a is not expressible as a = bc for some b, c ∈ R, then

(f ◦ g) (a) =∨

a=

n∑

i=1

bici

{

n∧

i=1

{f (bi) ∧ g (ci)}

}

= 0 ∧ 0 = 0.


Thus (f ◦ g)+ (a) = (f ◦ g) (a) ∨ 0.5 = 0.5. Since a is not expressible as a = bc,so

(

f+ ◦ g+)

(a) =∨

a=

n∑

i=1

bici

{

n∧

i=1

{

f+ (bi) ∧ g + (ci)}

}

=∨

a=

n∑

i=1

bici

{

n∧

i=1

{f (bi) ∧ g (ci) ∨ 0.5}

}

= 0.5.

Thus in this case (f ◦ g)+ = (f+ ◦ g+) . But if a is expressible as a = xy forsome x, y ∈ R. So,

(f ◦ g)+ (a) = (f ◦ g) (a) ∨ 0.5 =∨

a=

n∑

i=1

xiyi

{

n∧

i=1

{f (xi) ∧ g (yi)}

}

∨ 0.5

=∨

a=

n∑

i=1

xiyi

{

n∧

i=1

{(f (xi) ∨ 0.5) ∧ (g (yi) ∨ 0.5)}

}

=∨

a=

n∑

i=1

xiyi

{

f+ (xi) ∧ g+ (yi)}

=(

f+ ◦ g+)

(a) .

Definition 28. Let A be a non-empty subset of an LA-ring R. Then the

lower and upper parts of the characteristic function are C−A =

{

0.5 if a ∈ A0 if a /∈ A

and C+A =

{

1 if a ∈ A0.5 if a /∈ A

.

Theorem 29. If A and B are non-empty subsets of an LA-ring R, thenthe following properties hold.


(1) (CA ∧ CB)− = C−

A∩B

(2) (CA ∨ CB)− = C−

A∪B

(3) (CA ◦ CB)− = C−

AB


Theorem 30. The lower part of characteristic function C−L is an (∈,∈ ∨q)-

fuzzy left ideal of an LA–ring R if and only if L is a left ideal of R.

Proof. If L is a left ideal of R, then trivially C−L is an (∈,∈ ∨q)-fuzzy left

ideal of R.Conversely, assume that C−

L is an (∈,∈ ∨q)-fuzzy left ideal of R. If y ∈ L,then C−

L (y) = 0.5.and y0.5 ∈ C−L . Since C−

L is an (∈,∈ ∨q)-fuzzy left idealof R, so (xy)0.5 ∈ ∨qC−

L and (x− y)0.5 ∈ ∨qC−L , which further implies that

(xy)0.5 ∈ C−L or (xy)0.5 qC

−L and (x− y)0.5 ∈ C−

L or (x− y)0.5 qC−L . Hence

C−L (xy) ≥ 0.5 or C−

L (xy)+0.5 > 1 and C−L (x− y) ≥ 0.5 or C−

L (x− y)+0.5 >1. But C−

L (xy) + 0.5 > 1 and C−L (x− y) + 0.5 > 1 is not possible, because

C−L (xy) ≤ 0.5 and C−

L (x− y) ≤ 0.5. Thus C−L (xy) ≥ 0.5 and C−

L (x− y) ≥ 0.5,which implies that C−

L (xy) = 0.5 and C−L (x− y) = 0.5. Hence xy ∈ L and

x− y ∈ L. Thus L is a left ideal of R.

Similarly we can prove that the lower part of characteristic function C−R is an

(∈,∈ ∨q)-fuzzy right ideal of R if and only if R is a right ideal of R. Thus thelower part of characteristic function C−

I is an (∈,∈ ∨q)-fuzzy two-sided idealof R if and only if I is a two-sided ideal of R.

Theorem 31. If Q is a non empty subset of an LA-ring R, then Q is aquasi-ideal of R if and only if the lower part of characteristic function C−

Q is an(∈,∈ ∨q)-fuzzy quasi-ideal of R.


Theorem 32. If f is an (∈,∈ ∨q)-fuzzy left (right) ideal of an LA-ringR, then f− is a fuzzy left (right) ideal of R.

Proof. Let f be an (∈,∈ ∨q)-fuzzy left ideal of R. Then for all a, b ∈ R,we have f (ab) ≥ f (b) ∧ 0.5, and f (a− b) ≥ f (a) ∧ f (b) ∧ 0.5. This impliesthat f (ab) ∧ 0.5 ≥ f (b) ∧ 0.5 and f (a− b) ∧ 0.5 ≥ f (a) ∧ f (b) ∧ 0.5. Sof− (ab) ≥ f− (b) and f− (a− b) ≥ f− (a) ∧ f− (b) . Thus f− is a fuzzy leftideal of R. Similarly if f is an (∈,∈ ∨q)-fuzzy right ideal of R, then for alla, b ∈ R, we have f (ab) ≥ f (a) ∧ 0.5, and f (a− b) ≥ f (a) ∧ f (b) ∧ 0.5. Thisimplies that f (ab) ∧ 0.5 ≥ f (a) ∧ 0.5 and f (a− b) ∧ 0.5 ≥ f (a) ∧ f (b) ∧ 0.5.


So f− (ab) ≥ f− (a) and f− (a− b) ≥ f− (a) ∧ f− (b) . Thus f− is a fuzzy rightideal of R. Thus f− is a fuzzy right ideal of R.

Theorem 33. For an LA-ring R, the following conditions are equivalent.(1) R is regular.

(2) (f ∧ g)− = (f ◦ g)− for every (∈,∈ ∨q)-fuzzy right ideal f and every(∈,∈ ∨q)-fuzzy left ideal g of R.

Proof. (1) =⇒ (2) Let f be (∈,∈ ∨q)-fuzzy right ideal and g be (∈,∈ ∨q)-fuzzy left ideal of R. Now for all a ∈ R, we have

(f ◦ g)− (a) = (f ◦ g) (a) ∧ 0.5 =∨

a=

n∑

i=1

xiyi

{

n∧

i=1

{f (xi) ∧ g (yi)}

}

∧ 0.5

=∨

a=

n∑

i=1

xiyi

{

n∧

i=1

{(f (xi) ∧ 0.5) ∧ (g (yi) ∧ 0.5)} ∧ 0.5

}

≤∨

a=

n∑

i=1

xiyi

{{f (xiyi) ∧ g (xiyi)} ∧ 0.5} = f (a) ∧ g (a) ∧ 0.5

= (f ∧ g) (a) ∧ 0.5 = (f ∧ g)− (a) .

So (f ◦ g)− ≤ (f ∧ g)− . Since R is regular, so for each a ∈ R there exists anelement x ∈ R such that a = (ax) a. Thus

(f ∧ g)− (a) = (f ∧ g) (a) ∧ 0.5 ≤ (f (ax) ∧ g (a)) ∧ 0.5

≤∨

a=

n∑

i=1

yizi

{

n∧

i=1

{f (yi) ∧ g (zi)}

}

∧ 0.5

= (f ◦ g) (a) ∧ 0.5 = (f ◦ g)− (a) .

So (f ∧ g)− ≤ (f ◦ g)− . Thus (f ∧ g)− = (f ◦ g)− .(2) =⇒ (1) Let a ∈ R. Then L = aR is a left ideal of R and R = aR ∪ Ra

is a right ideal of R generated by a. Then by Theorem 30, the lower part


of characteristic functions C−R and C−

L of R and L are (∈,∈ ∨q)-fuzzy rightideal and (∈,∈ ∨q)-fuzzy left ideal of R, respectively. Thus we have C−

RL =(CR ◦ CL)

− = (CR ∧CL)− = C−

R∩L. Thus R ∩ L = RL. Hence it follows fromTheorem 33 that R is regular.

Theorem 34. If R is an LA-ring with left identity e such that (xe)R = xRfor all x ∈ R, then the following conditions are equivalent.

(1) R is regular.

(2) ((h ∧ f) ∧ g)− ≤ ((h ◦ f) ◦ g)− for every (∈,∈ ∨q)-fuzzy right ideal h,every (∈,∈ ∨q)-fuzzy generalized bi-ideal f and for every (∈,∈ ∨q)-fuzzy leftideal g of R.

(3) ((h ∧ f) ∧ g)− ≤ ((h ◦ f) ◦ g)− for every (∈,∈ ∨q)-fuzzy right ideal h,every (∈,∈ ∨q)-fuzzy bi-ideal f and for every (∈,∈ ∨q)-fuzzy left ideal g of R.

(4) ((h ∧ f) ∧ g)− ≤ ((h ◦ f) ◦ g)− for every (∈,∈ ∨q)-fuzzy right ideal h,every (∈,∈ ∨q)-fuzzy quasi bi-ideal f and for every (∈,∈ ∨q)-fuzzy right idealg of R.

Proof. (1) =⇒ (2) Let h, f and g be any (∈,∈ ∨q)-fuzzy right ideal, (∈,∈ ∨q)-fuzzy generalized bi-ideal and for every (∈,∈ ∨q)-fuzzy right ideal g of R re-spectively. Let a be any element of R. Since R is regular, so there exists anelement x ∈ R such that a = (ax) a.

Hence we have

((h ◦ f) ◦ g)− (a) =

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(h ◦ f) (yi) ∧ g (zi)}

}

∧ 0.5.

Now using medial law and the property (xe)R = xR for all x ∈ R, we havea = (ax) a = (ax) (ea) = (ae) (xa) = a (xa) .

This implies that

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(h ◦ f) (yi) ∧ g (zi)}

}

∧ 0.5


≥ (h ◦ f) (a) ∧ g (xa) ∧ 0.5

≥

∨

a=

n∑

i=1

piqi

{

n∧

i=1

{h (pi) ∧ f (qi)}

}

∧ (g (a) ∧ 0.5) ∧ 0.5

≥ (h (ax) ∧ f (a)) ∧ g (a) ∧ 0.5

≥ (h (a) ∧ 0.5 ∧ f (a)) ∧ g (a) ∧ 0.5

= (h (a) ∧ f (a)) ∧ g (a) ∧ 0.5 = ((h ∧ f) ∧ g)− (a) .

Thus, ((h ∧ f) ∧ g)− ≤ ((h ◦ f) ◦ g)− .

(2) =⇒ (3) =⇒ (4) Straightforward.

(4) =⇒ (1) Let h and g be any (∈,∈ ∨q)-fuzzy right ideal and any (∈,∈ ∨q)-fuzzy left ideal of R, respectively. Since δ is an (∈,∈ ∨q)-fuzzy quasi-ideal ofR, by the assumption, we have

(h ∧ g)− (a) = (h ∧ g) (a) ∧ 0.5 = ((h ∧ δ) ∧ g) (a) ∧ 0.5

= ((h ∧ δ) ∧ g)− (a) ≤ ((h ◦ δ) ◦ g)− (a)

= ((h ◦ δ) ◦ g) (a)

∧0.5

=

∨

a=

n∑

i=1

bici

{

n∧

i=1

{(h ◦ δ) (bi) ∧ g (ci)}

}

∧ 0.5

=

∨

a=

n∑

i=1

bici

n∧

i=1

∨

b=

n∑

i=1

piqi

{

n∧

i=1

{h (pi) ∧ δ (qi)} ∧ g (ci)

}

∧ 0.5


=

∨

a=

n∑

i=1

bici

n∧

i=1

∨

b=

n∑

i=1

piqi

{

n∧

i=1

{h (pi) ∧ 1} ∧ g (ci)

}

∧ 0.5

=

∨

a=

n∑

i=1

bici

n∧

i=1

∨

b=

n∑

i=1

piqi

{

n∧

i=1

h (pi) ∧ g (ci)

}

∧ 0.5

=

∨

a=

n∑

i=1

bici

n∧

i=1

∨

b=

n∑

i=1

piqi

{

n∧

i=1

{h (pi) ∧ 0.5} ∧ g (ci)

}

∧ 0.5

≤

∨

a=

n∑

i=1

bici

n∧

i=1

∨

b=

n∑

i=1

piqi

{

n∧

i=1

h (piqi) ∧ g (ci)

}

∧ 0.5

=

∨

a=

n∑

i=1

bici

{

n∧

i=1

{{h (bi) ∧ g (ci)}}

}

∧ 0.5

= (h ◦ g) (a) ∧ 0.5 = (h ◦ g)− (a) .

Thus it follows that (h ∧ g)− ≤ (h ◦ g)− for every (∈,∈ ∨q)-fuzzy right ideal hand every (∈,∈ ∨q)-fuzzy left ideal g of R. But (h ◦ g)− ≤ (h ∧ g)−, therefore(h ◦ g)− = (h ∧ g)− . Hence it follows from Theorem 33 that R is regular.


Theorem 35. If R is an LA-ring with left identity e such that (xe)R = xRfor all x ∈ R. then the following conditions are equivalent.

(1) R is regular.

(2) f− = ((f ◦ δ) ◦ f)− for every (∈,∈ ∨q)-fuzzy generalized bi-ideal f ofR.

(3) f− = ((f ◦ δ) ◦ f)− for every (∈,∈ ∨q)-fuzzy bi-ideal f of R.

(4) f− = ((f ◦ δ) ◦ f)− for every (∈,∈ ∨q)-fuzzy quasi-ideal f of R, whereδ is the fuzzy subset of R mapping every element of R on 1

Proof. (1) =⇒ (2) : Let f be an (∈,∈ ∨q)-fuzzy generalized bi-ideal of Rand let a be any element of R. Since R is regular, so there exists an elementx ∈ R such that a = (ax) a. Hence we have

((f ◦ δ) ◦ f)− (a) = ((f ◦ δ) ◦ f) (a) ∧ 0.5

=

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(f ◦ δ) (yi) ∧ f (zi)}

}

∧ 0.5

≥ {(f ◦ δ) (ax) ∧ f (a)} ∧ 0.5

=

∨

ax=

n∑

i=1

piqi

{

n∧

i=1

{f (pi) ∧ δ (qi)}

}

∧ f (a)

∧ 0.5

≥ {(f (a) ∧ δ (x)) ∧ f (a)} ∧ 0.5

= {(f (a) ∧ 1) ∧ f (a)} ∧ 0.5 = f (a) ∧ 0.5 = f− (a) .

Thus ((f ◦ δ) ◦ f)− ≥ f−. Since f is an (∈,∈ ∨q)-fuzzy generalized bi-ideal ofR. So we have

((f ◦ δ) ◦ f)− (a) = ((f ◦ δ) ◦ f) (a) ∧ 0.5


=

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(f ◦ δ) (yi) ∧ f (zi)}

}

∧ 0.5

=

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{

∨y=

n∑

i=1

piqi

{

n∧

i=1

{f (pi) ∧ δ (qi)}

}

∧ f (zi)

}}

∧ 0.5

=

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{

∨y=

n∑

i=1

piqi

{

n∧

i=1

{f (pi) ∧ 1}

}

∧ f (zi)

}}

∧ 0.5

=

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{

∨y=

n∑

i=1

piqi

n∧

i=1

f (pi) ∧ f (zi)

}}

∧ 0.5

≤∨

a=

n∑

i=1

(piqi)zi

{

n∧

i=1

f ((piqi) zi) ∧ 0.5

}

= f (a) ∧ 0.5 = f− (a) .

So, ((f ◦ δ) ◦ f)− ≤ f−. Thus ((f ◦ δ) ◦ f)− = f−.Now (2) =⇒ (3) =⇒ (4) trivially hold.(4) =⇒ (1) Let A be any quasi-ideal of R. We have

(AR)A ⊆ (AR)R ∩ (RR)A ⊆ AR ∩RA ⊆ A.

Let a be any element of A. Since CA is an (∈,∈ ∨q)-fuzzy quasi-ideal of R, wehave


∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(CA ◦ δ) (yi) ∧ CA (zi)}

}

∧ 0.5

= ((CA ◦ δ) ◦ CA) (a) ∧ 0.5

= ((CA ◦ δ) ◦ CA)− (a) = C−

A (a) = 0.5.

This implies that

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(CA ◦ δ) (yi) ∧ CA (zi)}

}

≥ 0.5.

But

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(CA ◦ δ) (yi) ∧ CA (zi)}

}

6= 0.5.

Thus

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(CA ◦ δ) (yi) ∧ CA (zi)}

}

> 0.5.

Hence

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(CA ◦ δ) (yi) ∧ CA (zi)}

}

= 1.

This implies that there exist elements b and c of R such that (CA ◦ δ) (b) = 1


and CA (c) = 1 with a = bc. Thus we have

∨

b=

n∑

i=1

piqi

{

n∧

i=1

{CA (pi) ∧ δ (qi)}

}

= (CA ◦ δ) (b) = 1.

This implies that there exist elements d and e of R such that CA (d) = 1 andδ (e) = 1 with b = de. Thus d, c ∈ A and e ∈ R and so a = bc = (de) c ∈ (AR)A.Therefore, A ⊆ (AR)A, and so A = (AR)A. Hence R is regular.

Theorem 36. If R is an LA-ring with left identity e such that (xe)R = xRfor all x ∈ R. then the following conditions are equivalent.

(1) R is regular.(2) (f ∧ g)− ≤ (f ◦ g)− for every (∈,∈ ∨q)-fuzzy quasi-ideal f and every

(∈,∈ ∨q)-fuzzy left ideal g of R.(3) (f ∧ g)− ≤ (f ◦ g)− for every (∈,∈ ∨q)-fuzzy bi-ideal f and every (∈,∈ ∨q)-

fuzzy left ideal g of R.(4) (f ∧ g)− ≤ (f ◦ g)− for every (∈,∈ ∨q)-fuzzy generalized bi-ideal f and

every (∈,∈ ∨q)-fuzzy left ideal g of R.

Proof. (1) =⇒ (4) Let f and g be any (∈,∈ ∨q)-fuzzy generalized bi-idealand any (∈,∈ ∨q)-fuzzy left ideal of R respectively. Let a be any element of R.Therefore there exist an element x ∈ R such that a = (ax) a. Thus we have

(f ◦ g)− (a) = (f ◦ g) (a) ∧ 0.5 =

∨

a=

n∑

i=1

yizi

{

n∧

i=1

f (yi) ∧ g (zi)

}

∧ 0.5.

Now, a = (ax) a = (ax) (ea) = (ae) (xa) = a (xa) because (xe)R = xR for allx ∈ R. It follows that

∨

a=

n∑

i=1

yizi

{

n∧

i=1

f (yi) ∧ g (zi)

}

∧ 0.5 ≥ f (a) ∧ g (xa) ∧ 0.5


≥ f (a) ∧ (g (a) ∧ 0.5) ∧ 0.5

= (f ∧ g) (a) ∧ 0.5

= (f ∧ g)− (a) .

So (f ◦ g)− ≥ (f ∧ g)− . (4) =⇒ (3) =⇒ (2) are obvious.(2) =⇒ (1) Let f be an (∈,∈ ∨q)-fuzzy right ideal and g be an (∈,∈ ∨q)-

fuzzy left ideal of R. Since every (∈,∈ ∨q)-fuzzy right ideal of R is an (∈,∈ ∨q)-fuzzy quasi-ideal of R. So (f ◦ g)− ≥ (f ∧ g)− . Now

(f ◦ g)− (a) = (f ◦ g) (a) ∧ 0.5 =

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{f (yi) ∧ g (zi)}

}

∧ 0.5

=

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{f (yi) ∧ g (zi)}

}

∧ 0.5

=

∨

a=

n∑

i=1

yizi

{

n∧

i=1

{(f (yi) ∧ 0.5) ∧ (g (zi) ∧ 0.5)}

}

∧ 0.5

≤∨

a=

n∑

i=1

yizi

({

n∧

i=1

{f (yizi) ∧ g (yizi)}

}

∧ 0.5

)

= f (a) ∧ g (a) ∧ 0.5

= (f ∧ g) (a) ∧ 0.5 = (f ∧ g)− (a) .So (f ◦ g)− ≤ (f ∧ g)− .

Hence (f ◦ g)− = (f ∧ g)− for every (∈,∈ ∨q)-fuzzy right ideal f of R, andevery (∈,∈ ∨q)-fuzzy left ideal g of R. Thus by Theorem 33, R is regular.


4. Conclusion

This paper has explored theoretical methods of evaluation to identify the blon-deness of the (∈,∈ ∨q)-fuzzy ideals. A functional approach was used to under-take a characterization of this structure leading to a determination of some in-teresting LA-ring theoretic properties of the generated structures. Similar typesof characterization in section 3, can easily be obtained for (∈,∈ ∨qk)-fuzzy ide-als as well as (∈γ ,∈γ ∨qδ)-fuzzy ideals. In future work, we will discuss differentclasses of LA-rings using (∈,∈ ∨qk)-fuzzy ideals as well as (∈γ ,∈γ ∨qδ)-fuzzyideals.

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