# Semigroups Characterized by Their Generalized Fuzzy Ideals

Authors: Madad Khan; Saima Anis
Journal: Journal of Mathematics
 (2013)

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## Abstract

We have characterized right weakly regular semigroups by the properties of their(∈,∈∨qk)-fuzzy ideals.


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## Body

## 1. Introduction

Usually the models of real world problems in almost all disciplines like in engineering, medical science, mathematics, physics, computer science, management sciences, operations research, and artificial intelligence are mostly full of complexities and consist of several types of uncertainties while dealing with them in several occasion. To overcome these difficulties of uncertainties, many theories had been developed such as rough sets theory, probability theory, fuzzy sets theory, theory of vague sets, theory of soft ideals, and the theory of intuitionistic fuzzy sets. Zadeh discovered the relationships of probability and fuzzy set theory in [1] which has appropriate approach to deal with uncertainties. Many authors have applied the fuzzy set theory to generalize the basic theories of Algebra. The concept of fuzzy sets in structure of groups was given by Rosenfeld [2]. The theory of fuzzy semigroups and fuzzy ideals in semigroups was introduced by Kuroki in [3, 4]. The theoretical exposition of fuzzy semigroups and their application in fuzzy coding, fuzzy finite state machines, and fuzzy languages was considered by Mordeson. The concept of belongingness of a fuzzy point to a fuzzy subset by using natural equivalence on a fuzzy subset was considered by Murali [5]. By using these ideas, Bhakat and Das [6, 7] gave the concept of (α,β)-fuzzy subgroups by using the “belongs to” relation ∈ and “quasi-coincident with” relation q between a fuzzy point and a fuzzy subgroup and introduced the concept of an (∈,∈∨q)-fuzzy subgroups, where α,β∈{∈,q,∈∨q,∈∧q} and α≠∈∧q. In particular, (∈,∈∨q)-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. These fuzzy subgroups are further studied in [8, 9]. The concept of (∈,∈∨qk)-fuzzy subgroups is a viable generalization of Rosenfeld’s fuzzy subgroups. Davvaz defined (∈,∈∨qk)-fuzzy subnearrings and ideals of a near ring in [10]. Jun and Song initiated the study of (α,β)-fuzzy interior ideals of a semigroup in [11] which is the generalization of fuzzy interior ideals [12]. In [13], Kazanci and Yamak studied (∈,∈∨qk)-fuzzy bi-ideals of a semigroup.In this paper we have characterized right regular semigroups by the properties of their right ideal, bi-ideal, generalized bi-ideal, and interior ideal. Moreover we characterized right regular semigroups in terms of their(∈,∈∨qk)-fuzzy right ideal, (∈,∈∨qk)-fuzzy bi-ideal, (∈,∈∨qk)-fuzzy generalized bi-ideal, (∈,∈∨qk)-fuzzy bi-ideal, and (∈,∈∨qk)-fuzzy interior ideals.Throughout this paperS denotes a semigroup. A nonempty subset A of S is called a subsemigroup of S if A2⊆A. A nonempty subset J of S is called a left (right) ideal of S if SJ⊆I(JS⊆I). J is called a two-sided ideal or simply an ideal of S if it is both left and right ideal of S. A nonempty subset B of S is called a generalized bi-ideal of S if BSB⊆B. A nonempty subset B of S is called a bi-ideal of S if it is both a subsemigroup and a generalized bi-ideal of S. A subsemigroup I of S is called an interior ideal of S if SIS⊆I.An semigroupS is called a right weakly regular if for every a∈S there exist x,y∈S such that a=axay.Definition 1.
For a fuzzy setf of a semigroup S and t∈(0,1], the crisp set U(f;t)={x∈S such that f(x)≥t} is called level subset of f.Definition 2.
A fuzzy subsetf of a semigroup S of the form
(1)f(y)={t∈(0,1]ify=x,0ify≠x
is said to be a fuzzy point with support x and value t and is denoted by xt.A fuzzy pointxt is said to belong to (resp., quasi-coincident with) a fuzzy set f, written as xt∈f(resp., xtqf), if f(x)≥t(resp., f(x)+t>1). If xt∈f or xtqf, then we write xt∈∨qf. The symbol ∈∨q¯ means ∈∨q does not hold. For any two fuzzy subsets f and g of S, f≤g means that, for all x∈S, f(x)≤g(x).Generalizing the concept ofxtqf, Jun [12, 14] defined xtqkf, where k∈[0,1), as f(x)+t+k>1. xt∈∨qkf if xt∈f or xtqkf.


## 2.(∈,∈∨qk)-Fuzzy Ideals in Semigroups

Definition 3.
A fuzzy subset ofS is called an (∈,∈∨qk)-fuzzy subsemigroup of S if for all x,y∈S and t,r∈(0,1] the following condition holds: xt∈f and yr∈f imply (xy)min{t,r}∈∨qkf.Lemma 4 (see [15]).
Letf be a fuzzy subset of S. Then f is an (∈,∈∨qk)-fuzzy subsemigroup of S if and only if f(xy)≥min{f(x),f(y),(1-k)/2}.Definition 5.
A fuzzy subsetf of S is called an (∈,∈∨qk)-fuzzy left (right) ideal of S if for all x,y∈S and t,r∈(0,1] the following condition holds: yr∈f implies (xy)t∈∨qkf(xt∈fimplies(xy)t∈∨qkf).Lemma 6 (see [15]).
Letf be a fuzzy subset of S. Then f is an (∈,∈∨qk)-fuzzy left (right) ideal of S if and only if f(xy)≥min{f(y),(1-k)/2}(f(xy)≥min{f(x),(1-k)/2}).Definition 7.
A fuzzy subsemigroupf of a semigroup S is called an (∈,∈∨qk)-fuzzy bi-ideal of S if for all x,y,z∈S and t,r∈(0,1] the following condition holds: xt∈f and zt∈f imply (xyz)t∈∨qkf.Lemma 8 (see [15]).
A fuzzy subsetf of S is an (∈,∈∨qk)-fuzzy bi-ideal of S if and only if it satisfies the following conditions:(i)
f(xy)≥min{f(x),f(y),(1-k)/2} for all x,y∈S and k∈[0,1);(ii)
f(xyz)≥min{f(x),f(z),(1-k)/2} for all x,y,z∈S and k∈[0,1).Definition 9.
A fuzzy subsetf of a semigroup S is called an (∈,∈∨qk)-fuzzy generalized bi-ideal of S if for all x,y,z∈S and t,r∈(0,1] the following condition holds: xt∈f and zt∈f imply (xyz)t∈∨qkf.Lemma 10 (see [15]).
A fuzzy subsetf of S is an (∈,∈∨qk)-fuzzy generalized bi-ideal of S if and only if f(xyz)≥min{f(x),f(z),(1-k)/2} for all x,y,z∈S and k∈[0,1).Definition 11.
A fuzzy subsemigroupf of a semigroup S is called an (∈,∈∨qk)-fuzzy interior ideal of S if for all x,y,z∈S and t,r∈(0,1] the following condition holds: yt∈f imply (xyz)t∈∨qkf.Lemma 12 (see [15]).
A fuzzy subsetf of S is an (∈,∈∨qk)-fuzzy interior ideal of S if and only if it satisfies the following condition:(i)
f(xy)≥min{f(x),f(y),(1-k)/2} for all x,y∈S and k∈[0,1);(ii)
f(xyz)≥min{f(y),(1-k)/2} for all x,y,z∈S and k∈[0,1).Example 13.
LetS={1,2,3,} be a semigroup with binary operation “·,” as defined in the following Cayley table:
(2)·123111122223333
Clearly(S,·) is regular semigroup and {1}, {2}, and {3} are left ideals of S. Let us define a fuzzy subset δ of S as
(3)δ(1)=0.9,δ(2)=0.6,δ(3)=0.5.
Then clearlyδ is an (∈,∈∨q)-fuzzy ideal of S.Lemma 14 (see [15]).
A nonempty subsetR of a semigroup S is right (left) ideal if and only if (CR)k is an (∈,∈∨qk)-fuzzy right (left) ideal of S.Lemma 15.
A nonempty subsetI of a semigroup S is an interior ideal if and only if (CI)k is an (∈,∈∨qk)-fuzzy interior ideal of S.Lemma 16.
A nonempty subsetB of a semigroup S is bi-ideal if and only if (CB)k is an (∈,∈∨qk)-fuzzy bi-ideal of S.Lemma 17.
Letf and g be any fuzzy subsets of semigroup S. Then following properties hold:(i)
(f∧kg)=(fk∧gk),(ii)
(f∘kg)=(fk∘gk).Proof.
It is straightforward.Lemma 18.
LetA and B be any nonempty subsets of a semigroup S. Then the following properties hold:(i)
(CA∧kCB)=(CA∩B)k,(ii)
(CA∘kCB)=(CAB)k.Proof.
It is straightforward.


## 3. Characterizations of Regular Semigroups

Theorem 19.
For a semigroupS, the following conditions are equivalent:(i)
S is regular;(ii)
L1∩L2∩B⊆BL1L2 for left ideals L1, L2, and bi-ideal B of a semigroup S.(iii)
L[a]∩L[a]∩B[a]⊆B[a]L[a]L[a], for some a in S;Proof.
(
i
)
⇒
(
ii
): Let S be regular semigroup, then for an element a∈S there exists x∈S such that a=axa. Let a∈L1∩L2∩B, where B is a bi-ideal and L1 and L2 are left ideals of S. So a∈L1, a∈L2, and a∈B.
Asa=axa=axaxa=axaxaxa∈(BSB)SL1SL2⊆BL1L2. Thus L1∩L2∩B⊆BL1L2.
(
ii
)
⇒
(
iii
) is obvious.
(
iii
)
⇒
(
i
): As a∪Sa and a∪a2∪aSa are left ideal and bi-ideal of S generated by a, respectively, thus by assumption we have
(4)(a∪Sa)∩(a∪Sa)∩(a2∪aSa)⊆(a∪a2∪aSa)(a∪Sa)(a∪Sa)⊆(a∪a2∪aSa)S(a∪Sa)⊆(a∪a2∪aSa)(a∪Sa)=a2∪aSa∪a3∪a2Sa∪aSa2∪aSaSa⊆a2∪a3∪aSa.
Thusa=a2=aa=a2a=aaa or a=a3=aaa or a=axa, for some x in S. Hence S is regular semigroup.Theorem 20.
For a semigroupS, the following conditions are equivalent:(i)
S is regular;(ii)
R∩L1∩L2⊆RL1L2  for every right ideal R and bi-ideal B of a semigroup S;(iii)
R[a]∩L[a]∩L[a]⊆R[a]L[a]L[a], for some a in S.Proof.
(
i
)
⇒
(
ii
): Let S be regular semigroup, then for an element a∈S there exists x∈S such that a=axa. Let a∈R∩L1∩L2, where R is right ideal and L1, and L2 are left ideals of S. So a∈R, a∈L1and a∈L2. As a=axaxa∈(RS)L1SL2⊆RL1L2. Thus R∩L1∩L2⊆RL1L2.
(
ii
)
⇒
(
iii
) is obvious.
(
iii
)
⇒
(
i
): As a∪aS is right ideal and a∪Sa is left ideal of S generated by a, respectively, thus by assumption we have
(5)(a∪Sa)∩(a∪aS)∩(a∪aS)⊆(a∪aS)(a∪Sa)(a∪Sa)⊆(a∪aS)S(a∪Sa)⊆(a∪aS)(a∪Sa)=a2∪aSa∪aSa∪aSSa⊆a2∪aSa.
Thusa=a2 or a=axa, for some x in S. Hence S is regular semigroup.Theorem 21.
For a semigroupS, the following conditions are equivalent:(i)
S is regular;(ii)
f∘kg∘kh≥f∧kg∧kh for every (∈,∈∨qk)-fuzzy right ideal f, (∈,∈∨qk)-fuzzy left ideals g, and h of a semigroup S.Proof.
(
i
)
⇒
(
ii
): Let f be (∈,∈∨qk)-fuzzy right ideal, g and h any (∈,∈∨qk)-fuzzy left ideals of S. Since S is regular, therefore for each a∈S there exists x∈S such that
(6)a=axa=axaxa.
Thus(7)(f∘kg∘kh)(a)=(f∘g∘h)(a)∧1-k2=(a=pq{f(p)∧(g∘h)(q)})∧1-k2≥f(a)∧(g∘h)((xa)(xa))∧1-k2≥f(a)∧((xa)(xa)=bc{g(b)∧h(c)})∧1-k2≥f(a)∧g(xa)∧h(xa)∧1-k2≥f(a)∧g(a)∧h(a)∧1-k2≥f(a)∧g(a)∧h(a)∧1-k2.
(
ii
)
⇒
(
i
): Let R[a] be right ideal, and let L1[a] and L2[a] be any two left ideals of S generated by a, respectively.
Then(CR[a])k is any (∈,∈∨qk)-fuzzy right ideal, and (CL1[a])k and (CL2[a])k are any (∈,∈∨qk)-fuzzy left ideals of semigroup S, respectively. Let a∈S and b∈R[a]∩L1[a]∩L2[a]. Then b∈R[a], b∈L1[a], and b∈L2[a]. Now
(8)1-k2≤(CR[a]∩L1[a]∩L2[a])k(b)=((CR[a])k∧k(CL1[a])k∧k(CL2[a])k)(b)≤((CR[a])k∘k(CL1[a])k∘k(CL2[a])k)(b)=(CR[a]L1[a]L2[a])k(b).
Thusb∈R[a]L1[a]L2[a]. Therefore R[a]∩L1[a]∩L2[a]⊆R[a]L1[a]L2[a].
So by Theorem20, S is regular.


## 4. Characterizations of Right Weakly Regular Semigroups in Terms of(∈,∈∨qk)-Fuzzy Ideals

Theorem 22.
For a semigroupS, the following conditions are equivalent:(i)
S is right weakly regular;(ii)
R∩L∩I⊆RLI for every right ideal, left ideal, and interior ideal of S, respectively;(iii)
R[a]∩L[a]∩I[a]⊆R[a]L[a]I[a].Proof.
(
i
)
⇒
(
ii
): Let S be right weakly regular semigroup, and let R, L, and I be right ideal, left ideal, and interior ideal of S, respectively. Let a∈R∩L∩I then a∈R, a∈L, and a∈I. Since S is right weakly regular semigroup so for a there exist x,y∈S such that
(9)a=axay=axaxayy=axaxaxayyy∈(RS)(SL)(SIS)⊆RLI.
Thereforea∈RLI. So R∩L∩I⊆RLI.
(
ii
)
⇒
(
iii
) is obvious.
(
iii
)
⇒
(
i
): As a∪aS, a∪Sa, and a∪a2∪SaS are right ideal, left ideal, and interior ideal of S generated by an element a of S, respectively, thus by assumption, we have
(10)(a∪aS)∩(a∪Sa)∩(a∪a2∪SaS)⊆(a∪aS)(a∪Sa)(a∪a2∪SaS)=(a2∪aSa∪aSa∪aSSa)(a∪a2∪SaS)=a3∪a4∪a2SaS∪aSa2∪aSa3∪aSaSaS∪aSSa2∪aSSa3∪aSSaSaS⊆a3∪a4∪aSa∪aSaS.
Thusa=a3=aaa=aaa3=aaa2a or a=axa=axaxa or a=auav, for some x,u,v in S. Hence S is right weakly regular semigroup.Theorem 23.
For a semigroupS, the following conditions are equivalent:(i)
S is right weakly regular;(ii)
f∧g∧h≤f∘g∘h for every fuzzy right ideal, fuzzy left ideal, and fuzzy interior ideal of S, respectively.Proof.
(
i
)
⇒
(
ii
): Let f,  g, and h be any (∈,∈∨qk)-fuzzy right ideal, (∈,∈∨qk)-fuzzy generalized bi-ideal, and (∈,∈∨qk)-fuzzy interior ideal of S. Since S is right weakly regular therefore for each a∈S there exist x,y∈S such that
(11)a=axay=(axay)(xay)=(ax)(ay)(xay)=(ax)(axayy)(xay)=(ax)(axa)(yyxay).
Then(12)(f∘kg∘kh)(a)=(f∘g∘h)(a)∧1-k2=(a=pq{f(p)∧(g∘h)(q)})∧1-k2≥f(ax)∧(g∘h)((axa)(yyxay))∧1-k2≥f(a)∧((axa)(yyxay)=bc{g(b)∧h(c)})∧1-k2≥f(a)∧g(axa)∧h(yyxay)∧1-k2≥f(a)∧g(a)∧h(a)∧1-k2≥f(a)∧g(a)∧h(a)∧1-k2.
Thereforef∧kg∧kh≤f∘kg∘kh.
Now(ii)⇒(i)
(
ii
)
⇒
(
i
): Let R[a], L[a], and I[a] be right ideal, left ideal, and interior ideal of S generated by a, respectively.
Then(CR[a])k, (CL[a])k, and (CI[a])k are (∈,∈∨qk)-fuzzy right ideal, (∈,∈∨qk)-fuzzy left ideal, and (∈,∈∨qk)-fuzzy interior ideal of semigroup S. Let a∈S and b∈R[a]∩L[a]∩I[a]. Then b∈R[a], b∈L[a], and b∈I[a]. Now
(13)1-k2≤(CR[a]∩L[a]∩I[a])k(b)=((CR[a])k∧k(CL[a])k∧k(CI[a])k)(b)≤((CR[a])k∘k(CL[a])k∘k(CI[a])k)(b)=(CR[a]L[a]I[a])k(b).
Thusb∈R[a]L[a]I[a]. Therefore R[a]∩L[a]∩I[a]⊆R[a]L[a]I[a]. Hence by Theorem 22,S is right weakly regular semigroup.Theorem 24.
For a semigroupS, the following conditions are equivalent:(i)
S is right weakly regular;(ii)
B∩L∩I⊆BLI for every bi-ideal, left ideal, and interior ideal of S, respectively;(iii)
B[a]∩L[a]∩I[a]⊆B[a]L[a]I[a].Proof.
(
i
)
⇒
(
ii
): Let S be right weakly regular semigroup, and B, L, and I be bi-ideal, left ideal, and interior ideal of S, respectively. Let a∈B∩L∩I then a∈B, a∈L, and a∈I. Since S is right weakly regular semigroup so for a there exist x,y∈S such that
(14)a=axay=axaxaxayyy=(axa)(xa)(xayyy)∈(BS)(SL)(SIS)⊆BLI.
Thereforea∈BLI.So B∩L∩I⊆BLI.
(
ii
)
⇒
(
iii
) is obvious.
(
iii
)
⇒
(
i
): As a∪a2∪aSa, a∪Sa, and a∪a2∪SaS are bi-ideal, left ideal, and interior ideal of S generated by an element a of S, respectively, thus by assumption we have
(15)(a∪a2∪aSa)∩(a∪Sa)∩(a∪a2∪SaS)⊆(a∪a2∪aSa)(a∪Sa)(a∪a2∪SaS)=(a2∪aSa∪a3∪a2Sa∪aSa2∪aSaSa)×(a∪a2∪SaS)=a3∪a4∪a2SaS∪aSa2∪aSa3∪aSaSaS∪a4∪a5∪a3SaS∪a2Sa2∪a2Sa3∪a2SaSaS∪aSa3∪aSa4∪aSa2SaSaSaSa2∪aSaSa3∪aSaSaSaS⊆a3∪a4∪a5∪aSa∪aSaS.
Thusa=a4=aaaa or a=a3=aaa=aaa3=aaa2a or a=axa=axaxa or a=auav, for some x,u,v in S. Hence S is right weakly regular semigroup.Theorem 25.
For a semigroupS, the following conditions are equivalent:(i)
S is right weakly regular;(ii)
f∧g∧h≤f∘g∘h for every fuzzy bi-ideal, fuzzy left ideal and fuzzy interior ideal of S, respectively;(iii)
f∧g∧h≤f∘g∘h for every fuzzy generalized bi-ideal, fuzzy left ideal, and fuzzy interior ideal of S, respectively.Proof.
(
i
)
⇒
(
iii
): Let f,  g, and h be any (∈,∈∨qk)-fuzzy generalized bi-ideal, (∈,∈∨qk)-fuzzy left ideal, and (∈,∈∨qk)-fuzzy interior ideal of S. Since S is right weakly regular for each a∈S there exist x,y∈S such that
(16)a=axay=(axay)(xay)=(ax)(ay)(xay)=(ax)(axayy)(xay)=(axa)(xa)(yyxay).
Then(17)(f∘kg∘kh)(a)=(f∘g∘h)(a)∧1-k2=(a=pq{f(p)∧(g∘h)(q)})∧1-k2≥f(axa)∧(g∘h)((xa)(yyxay))∧1-k2≥f(a)∧((xa)(yyxay)=bc{g(b)∧h(c)})∧1-k2≥f(a)∧g(xa)∧h(yyxay)∧1-k2≥f(a)∧g(a)∧h(a)∧1-k2≥f(a)∧g(a)∧h(a)∧1-k2.
Thereforef∧kg∧kh≤f∘kg∘kh.
(
iii
)
⇒
(
ii
) is obvious.
(
ii
)
⇒
(
i
): Let B[a], L[a], and I[a] be bi-ideal, left ideal, and interior ideal of S generated by a, respectively.
Then(CB[a])k, (CL[a])k, and (CI[a])k are (∈,∈∨qk)-fuzzy bi-ideal, (∈,∈∨qk)-fuzzy left ideal, and (∈,∈∨qk)-fuzzy interior ideal of semigroup S. Let a∈S and b∈B[a]∩L[a]∩I[a]. Then b∈B[a], b∈L[a], and b∈I[a]. Now
(18)1-k2≤(CB[a]∩L[a]∩I[a])k(b)=((CB[a])k∧k(CL[a])k∧k(CI[a])k)(b)≤((CB[a])k∘k(CL[a])k∘k(CI[a])k)(b)=(CB[a]L[a]I[a])k(b).
Thusb∈B[a]L[a]I[a]. Therefore B[a]∩L[a]∩I[a]⊆B[a]L[a]I[a]. Hence by Theorem 24, S is right weakly regular semigroup.Theorem 26.
For a semigroupS, the following conditions are equivalent:(i)
S is right weakly regular;(ii)
Q∩L∩I⊆QLI for every quasi-ideal Q, left ideal L, and interior ideal I of S, respectively;(iii)
Q[a]∩L[a]∩I[a]⊆Q[a]L[a]I[a].Proof.
(
i
)
⇒
(
ii
): Let S be right weakly regular semigroup, and let Q, L, and I be quasi-ideal, left ideal, and interior ideal of S, respectively. Let a∈Q∩L∩I then a∈Q, a∈L, and a∈I. Since S is right weakly regular semigroup so for a there exist x,y∈S such that
(19)a=axay=(axay)(xay)=a(xa)(yxay)∈Q(SL)(SIS)⊆QLI.
Thereforea∈QLI. So Q∩L∩I⊆QLI.
(
ii
)
⇒
(
iii
) is obvious.
(
i
i
i
)
⇒
(
i
): As a∪(aS∩Sa), a∪Sa, and a∪a2∪SaS are quasi-ideal, left ideal, and interior ideal of S generated by an element a of S, respectively, thus by assumption we have
(20)(a∪(aS∩Sa))∩(a∪Sa)∩(a∪a2∪SaS)⊆(a∪(aS∩Sa))(a∪Sa)(a∪a2∪SaS)⊆(a∪aS)(a∪Sa)(a∪a2∪SaS)=(a2∪aSa∪aSa∪aSSa)(a∪a2∪SaS)=a3∪a4∪a2SaS∪aSa2∪aSa3∪aSaSaS∪aSSa2∪aSSa3∪aSSaSaS⊆a3∪a4∪aSa∪aSaS.
Thusa=a4=aaaa or a=a3=aaa=aaa3=aaa2a or a=axa=axaxa or a=auav, for some x,u,v in S. Hence S is right weakly regular semigroup.Theorem 27.
For a semigroupS, the following conditions are equivalent:(i)
S is right weakly regular;(ii)
f∧g∧h≤f∘g∘h for every fuzzy quasi-ideal, fuzzy left ideal, and fuzzy interior ideal of S, respectively.Proof.
(
i
)
⇒
(
iii
): Let f,  g, and h be any (∈,∈∨qk)-fuzzy quasi-ideal, (∈,∈∨qk)-fuzzy left ideal, and (∈,∈∨qk)-fuzzy interior ideal of S. Since S is right weakly regular therefore for each a∈S there exist x,y∈S such that
(21)a=axay=(axay)(xay)=a(xa)(yxay).
Then(22)(f∘kg∘kh)(a)=(f∘g∘h)(a)∧1-k2=(a=pq{f(p)∧(g∘h)(q)})∧1-k2≥f(a)∧(g∘h)((xa)(yxay))∧1-k2≥f(a)∧((xa)(yxay)=bc{g(b)∧h(c)})∧1-k2≥f(a)∧g(xa)∧h(yxay)∧1-k2≥f(a)∧g(a)∧h(a)∧1-k2≥f(a)∧g(a)∧h(a)∧1-k2.
Thereforef∧kg∧kh≤f∘kg∘kh.
(
iii
)
⇒
(
ii
) is obvious.
(
ii
)
⇒
(
i
): Let Q[a], L[a], and I[a] be quasi-ideal, left ideal and interior ideal of S generated by a, respectively.
Then(CQ[a])k, (CL[a])k, and (CI[a])k are (∈,∈∨qk)-fuzzy quasi-ideal, (∈,∈∨qk)-fuzzy left ideal, and (∈,∈∨qk)-fuzzy interior ideal of semigroup S. Let a∈S and let b∈Q[a]∩L[a]∩I[a]. Then b∈Q[a], b∈L[a], and b∈I[a]. Now
(23)1-k2≤(CQ[a]∩L[a]∩I[a])k(b)=((CQ[a])k∧k(CL[a])k∧k(CI[a])k)(b)≤((CQ[a])k∘k(CL[a])k∘k(CI[a])k)(b)=(CQ[a]L[a]I[a])k(b).
Thusb∈Q[a]L[a]I[a]. Therefore Q[a]∩L[a]∩I[a]⊆Q[a]L[a]I[a]. Hence by Theorem 26, S is right weakly regular semigroup.



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*Source: 592708-2013-03-25.xml*