# The Lattices of Group Fuzzy Congruences and Normal Fuzzy Subsemigroups onE-Inversive Semigroups

Authors: Shoufeng Wang
Journal: The Scientific World Journal
 (2014)

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## Abstract

The aim of this paper is to investigate the lattices of group fuzzy congruences and normal fuzzy subsemigroups onE-inversive semigroups. We prove that group fuzzy congruences and normal fuzzy subsemigroups determined each other in E-inversive semigroups. Moreover, we show that the set of group t-fuzzy congruences and the set of normal subsemigroups with tip t in a given E-inversive semigroup form two mutually isomorphic modular lattices for every t∈0,1.


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## Body

## 1. Introduction

The investigation of fuzzy sets is initiated by Zadeh in [1]. As special fuzzy sets, fuzzy congruences on groups and semigroups have been extensively studied by many authors. In 1992, Kuroki [2] introduced fuzzy congruences on a group and characterized fuzzy congruences on a group in terms of fuzzy normal subgroups. In 1993, Samhan [3] studied the modularity condition in the fuzzy congruence lattice of a semigroup and derived that the fuzzy congruence lattice of a group is modular. In the same year, Al-Thukair [4] described the fuzzy congruences of an inverse semigroup and obtained a one-one correspondence between fuzzy congruence pairs and fuzzy congruences on an inverse semigroup. Moreover, Kuroki also studied the fuzzy congruences on inverse semigroups in [5] in which the notion of group congruences of a semigroup is provided. Das [6] considered the lattice of fuzzy congruences in an inverse semigroup by kernel-trace approaches. In 1995, Ajmal and Thomas considered the lattice structures of fuzzy congruences on a group and the lattice structures of fuzzy subgroups and fuzzy normal subgroups in a group in [7] and proved that the lattice of fuzzy normal subgroups of a group is modular in [8]. In 1997, Kim and Bae [9] studied the fuzzy congruences of groups and obtained several results which are analogs of some basic theorems of group theory. Also, Xie [10] studied the so-called fuzzy Rees congruences on semigroups in 1999.Several authors investigated fuzzy congruences for some special classes of semigroups. In 2000, Zhang [11] characterized the group fuzzy congruences on a regular semigroup by some fuzzy subsemigroups. In 2001, Tan [12] investigated fuzzy congruences of regular semigroups and proved that the lattice of fuzzy congruences on a regular semigroup is a disjoint union of some modular sublattices of the lattice. Recently, Li and Liu [13] characterized fuzzy good congruences of left semiperfect abundant semigroups and obtained sufficient and necessary conditions for an abundant semigroup to be left semiperfect.The class ofE-inversive semigroups is a very wide class of semigroups which contains groups, inverse semigroups, and regular semigroups as proper subclasses and some kinds of crisp congruences on this class of semigroups have been investigated extensively; see [14, 15] for example. In particular, Gigoń [14] considered the lattice of group crisp congruences on an E-inversive semigroup and proved that this lattice is modular. Inspired by the above facts, it is natural to study the fuzzy congruences on E-inversive semigroups. In fact, [16] has done some works in this aspect.In this paper, we shall investigate the lattices of group fuzzy congruences and normal fuzzy subsemigroups on anE-inversive semigroup. The notions of group t-fuzzy congruences and normal fuzzy subsemigroups with tip t on E-inversive semigroups are proposed and some properties of them are given. In particular, for a given E-inversive semigroup S, we prove that for any t∈[0,1] the set of group t-fuzzy congruences and the set of normal fuzzy subsemigroups with tip t on S form two mutually isomorphic modular lattices. Our results generalize and enrich several results obtained in [2, 3, 8, 9, 11, 14]. Notations and terminologies not given in this paper can be found in [17–19].


## 2. Preliminaries

A binary relation “≤” defined on a set A is apartial order on the set A if the following conditions hold identically in A:(
1
)  a≤a; (2)  a≤b and b≤a imply a=b; (3)  a≤b and b≤c imply a≤c.LetA be a subset of a poset (P,≤). An element p in P is an upper bound for A if a≤p for every a in A. An element p in P is thesupremum of A if p is an upper bound of A and p is the smallest among the upper bounds of A. Dually, we can define theinfimum of A. We denote the supremum and the infimum of A by ⋁a∈Aa and ⋀a∈Aa, respectively.A poset(L,≤) is called alattice if for every a,b in L both the supremum a∨b and the infimum a∧b of {a,b} exist in L. Amodular lattice is any lattice L which satisfies themodular law: x≤y implies that x∨(y∧z)=y∧(x∨z) for all x,y,z in L.Two latticesL1 and L2 areisomorphic if there is a bijection α from L1 to L2 such that for every a,b in L1 the following two equations hold: α(a∨b)=α(a)∨α(b) and α(a∧b)=α(a)∧α(b). If P1 and P2 are two posets and α is a map from P1 to P2, then we say α isorder-preserving if α(a)≤α(b) holds in P2 whenever a≤b holds in P1.On the theory of lattices, we need the following results.Lemma 1 (Theorem 2.3 of Chapter 1 in [17]).
Two latticesL1 and L2 are isomorphic if and only if there is a bijection α from L1 to L2 such that both α and α-1 are order-preserving.Lemma 2 (Theorem 4.2 of Chapter 1 in [17]).
LetP be a poset such that it has the largest element and the infimum of every nonempty subset exists. Then P is a lattice.Zadeh [1] defined a fuzzy subset μ in a set S as a mapping from S to the closed unit interval [0,1]. A fuzzy set μ in a set S is said to be contained in a fuzzy set η if μ(x)≤η(x) for all x in S and this is denoted by μ⊆η. The union μ∪η and the intersection μ∩η of two fuzzy sets μ and η in a set S are defined by
(1)μ∪η(x)=max⁡⁡(μ(x),η(x))=μ(x)∨η(x),μ∩η(x)=min⁡⁡(μ(x),η(x))=μ(x)∧η(x)
for all x in S. Further, if μi is a fuzzy subset in S for i∈I where I is an index set, then ⋂i∈I‍μi is defined by
(2)⋂i∈I‍μi(x)=inf⁡⁡{μi(x)∣i∈I}=⋀i∈Iμi(x)
for all x∈S.Asemigroup is a nonempty set with an associative binary operation. A semigroup S is calledE-inversive if for all a∈S there exists a′∈S such that a′aa′=a. In this case, we denote W(a)={a′∈S∣a′aa′=a′} and call the elements in W(a) theweak inverses of a for any a∈S. It is easy to see that groups, regular semigroups, and semigroups with zeros are all E-inversive semigroups. For more details on E-inversive semigroups, see [14, 15] and their references. Throughout this paper, we always assume that S is an E-inversive semigroup and let
(3)E(S)={e∈S∣e2=e}.Now, we give the concept oft-fuzzy congruences.Definition 3 (see [19]).
Lett∈[0,1]. A t-fuzzy equivalence on S is a fuzzy subset in S×S which satisfies the following conditions: (1)
(∀a∈S)ρ(a,a)=t,(2)
(∀a,b∈S)ρ(a,b)=ρ(b,a)≤t,(3)
(∀a,b,c∈S)ρ(a,b)≥ρ(a,c)∧ρ(c,b).A t-fuzzy equivalence ρ on S is called a t-fuzzycongruence if (4)
(∀a,b,c∈S)ρ(a,b)≤ρ(ac,bc)∧ρ(ca,cb),or, equivalently, (4′)
(∀a,b,c,d∈S)ρ(ab,cd)≥ρ(a,c)∧ρ(b,d).Similar to the proofs of Lemmas 2.2 and 2.3 in Kuroki [5], we have the following results.Lemma 4.
Letρ be a t-fuzzy congruence on S. For any a∈S, define a fuzzy subset ρa in S as follows: ρa(x)=ρ(a,x) for all x∈S.(1)
ρa=ρb if and only if ρ(a,b)=t for all a,b∈S.(2)
S/ρ={ρa∣a∈S} is a semigroup with the multiplication ρaρb=ρab for any a,b∈S.


## 3. Group Fuzzy Congruences

In this section, we consider some basic properties of group fuzzy congruences onS. In particular, we show that the set of group t-fuzzy congruences on S forms a modular lattice. Firstly, we give the concept of group t-fuzzy congruences which is parallel to that of usual group fuzzy congruences defined in Kuroki [5].Definition 5.
At-fuzzy congruence ρ on S is called a group t-fuzzy  congruence if the semigroup S/ρ is a group. One denotes the set of group t-fuzzy congruences on S by GFCt(S) and denotes GFC(S)=⋃t∈[0,1]GFCt(S).The following result provides a characterization of groupt-fuzzy congruences on S.Proposition 6.
At-fuzzy congruence ρ on S is a group t-fuzzy congruence if and only if (1)
(∀e,f∈E(S))ρ(e,f)=t;(2)
(∀a∈S)(∀a′∈W(a))ρ(aa′a,a)=t.Proof.
Ifρ is a group t-fuzzy congruence, then S/ρ is a group and so ρe is the identity of S/ρ for every e∈E(S). This implies that ρ(e,f)=t for all e,f∈E(S) by Lemma 4. Furthermore, since a′a∈E(S) for any a∈S and a′∈W(a), it follows that ρa′a is the identity of S/ρ and so ρa=ρaρa′a=ρaa′a. This yields that ρ(aa′a,a)=t by Lemma 4 again.
Conversely, leta∈S, a′∈W(a), and e∈E(S). Then by condition (1) and Lemma 4, ρaa′=ρa′a=ρe. By condition (2) and Lemma 4, we have
(4)ρa=ρaa′a=ρaρa′a=ρaρe,ρaρa′=ρaa′=ρe,
which implies that S/ρ is a group.Proposition 7.
Letρ∈GFCt(S), a,b∈S, and e∈E(S). Then
(5)ρ(a,b)=ρ(a,be)=ρ(ae,b)=ρ(a,eb)=ρ(ea,b).Proof.
Sinceρ∈GFCt(S), by Proposition 6, we have
(6)ρ(b,bb′b)=ρ(a,aa′a)=t=ρ(aa′,e)=ρ(e,b′b)
for all a′∈W(a) and b′∈W(b). This implies that
(7)ρ(a,be)≥ρ(a,aa′a)∧ρ(aa′a,be)=ρ(aa′a,be)≥ρ(a,b)∧ρ(aa′,e)=ρ(a,b).
On the other hand,
(8)ρ(a,b)≥ρ(a,be)∧ρ(be,bb′b)∧ρ(bb′b,b)≥ρ(a,be)∧ρ(be,bb′b)≥ρ(a,be)∧ρ(b,b)∧ρ(e,b′b)=ρ(a,be).
Therefore, ρ(a,b)=ρ(a,be). By similar arguments, we can show
(9)ρ(a,b)=ρ(ae,b)=ρ(a,eb)=ρ(ea,b)
for all a,b∈S and e∈E(S).As usual, forσ,τ∈GFCt(S), we define σ∘τ as follows:
(10)σ∘τ(a,b)=⋁x∈S(σ(a,x)∧τ(x,b))
for all a,b∈S. Then we have the following.Proposition 8.
Letσ,τ∈GFCt(S). (1)
σ∘τ=τ∘σ.(2)
σ∘τis the least groupt-fuzzy congruence ofScontainingσandτ.(3)
σ∩τis the greatest groupt-fuzzy congruence ofScontained inσandτ.Proof.
(
1
) For all a,b∈S and c′∈W(c), by Proposition 7, we have
(11)σ∘τ(a,b)=⋁x∈S(σ(a,x)∧τ(x,b))≥⋁c∈S(σ(a,bc′a)∧τ(bc′a,b))=⋁c∈S(τ(bc′a,b)∧σ(a,bc′a))=⋁c∈S(τ(bc′a,bc′c)∧σ(cc′a,bc′a))≥⋁c∈S(τ(a,c)∧σ(c,b))=τ∘σ(a,b).
By symmetry, we have σ∘τ(a,b)=τ∘σ(a,b) for all a,b∈S.
(
2
) Let a,b,c∈S. Since
(12)t=σ(a,a)∧τ(a,a)≤⋁x∈S(σ(a,x)∧τ(x,a))≤⋁x∈S(t∧t)=t,
we have σ∘τ(a,a)=t. Similarly, we have
(13)σ∘τ(a,b)=⋁x∈S(σ(a,x)∧τ(x,b))≤⋁x∈S(t∧t)=t.
In view of the proofs of Propositions 1.8 and 1.9 in Kim and Bae [9], σ∘τ is the least t-fuzzy congruence on S containing σ and τ. Finally, we can easily show that σ∘τ∈GFCt(S) by Proposition 6. This implies that σ∘τ is the least group t-fuzzy congruence of S containing σ and τ.
(
3
) This is clear.By Proposition8 and the proof of Theorem 1.12 in Kim and Bae [9], we have the following result.Theorem 9.
(
G
F
C
t
(
S
)
,
⊆
) forms a modular lattice for any t in [0,1].In the end of this section, we give some properties of groupt-fuzzy congruences on S which will be used in the final section.Proposition 10.
Letρ∈GFCt(S) and a,b∈S. (1)
ρ(aa′,a)=ρ(aa*,a)for alla′,a*∈W(a).(2)
ρ(a′b(a′b)′,a′b)=ρ(a,b)for alla′∈W(a)and(a′b)′∈W(a′b).Proof.
(
1
) Since ρ∈GFCt(S), by Proposition 6, we have
(14)ρ(aa′a,a)=ρ(aa*a,a)=t=ρ(aa*,aa′)=ρ(a′a,a*a).
This yields that ρ(aa′aa*,aa*)≥ρ(aa′a,a)=t whence ρ(aa′aa*,aa*)=t. Thus,
(15)ρ(aa*,a)≤ρ(aa′aa*,aa′a)=ρ(aa′a,aa′aa*)∧ρ(aa′aa*,aa*)≤ρ(aa′a,aa*)∧ρ(aa*,aa′)≤ρ(aa′a,aa′)=ρ(aa′,aa′a)∧ρ(aa′a,a)≤ρ(aa′,a).
By dual arguments, we can obtain that ρ(aa′,a)=ρ(aa*,a).
(2) In view of the fact thata′b(a′b)′,aa′∈E(S), by Proposition 7, we have
(16)ρ(a′b(a′b)′,a′b)≤ρ(aa′b(a′b)′,aa′b)=ρ(a,aa′b)=ρ(a,b)=ρ(aa′b(a′b)′,b)≤ρ(a′aa′b(a′b)′,a′b)=ρ(a′b(a′b)′,a′b).
Thus, ρ(a′b(a′b)′,a′b)=ρ(a,b).


## 4. Normal Fuzzy Subsemigroups

In this section, we consider some basic properties of normal fuzzy subsemigroups ofE-inverse semigroups.Definition 11.
Lett∈[0,1]. A fuzzy subset μ in S is called anormal  fuzzy  subsemigroup  with  tipt in S if (1)
(∀x,y∈S)μ(xy)≥μ(x)∧μ(y),(2)
(∀a,x,y∈S)t≥μ(a)≥μ(xay)∧μ(xy),(3)
(∀e∈E(S))μ(e)=t.We denote the set of normal fuzzy subsemigroups with tipt in S by NFSt(S) and let NFS(S)=⋃t∈[0,1]NFSt(S).Remark 12.
In fact, normal fuzzy subsemigroups with tip1 are introduced in Zhang [11] where this class of fuzzy subsemigroups is called complete inner-unitary subsemigroups.LetG be a group with identity e, t∈[0,1] and let μ be a fuzzy set in G. From Ajmal and Thomas [8], μ is called anormal fuzzy subgroup of G with tip t if for all x,y∈G the following conditions hold:
(17)μ(xy)≥μ(x)∧μ(y),μ(x-1)=μ(x),μ(xy)=μ(yx),μ(e)=t.
We assert that normal fuzzy subsemigroups with tip t are generalizations of normal fuzzy subgroups with tip t in the range of E-inversive semigroups. To see this, we need the following result.Proposition 13.
Letμ∈NFSt(S), a,b∈S, and a′∈W(a). (1)
μ(a)=μ(a′).(2)
μ(a)=μ(aa′a).(3)
μ(ab)=μ(ba).Proof.
(
1
) On the one hand, we have aa′∈E(S) and
(18)μ(a)≥μ(a′aa′)∧μ((a′)2)=μ(a′)∧μ((a′)2)=μ(a′).
On the other hand, since a′a∈E(S), we have μ(a′a)=t≥μ(a). This implies that
(19)μ(a′)≥μ(aa′a)∧μ(a2)≥(μ(a)∧μ(a′a))∧μ(a2)=μ(a).
Therefore, μ(a)=μ(a′).
(
2
) This follows from item (1) and the fact that aa′a∈W(a′).
(
3
) The result follows from Proposition 2.6 in Zhang [11].The following result justifies the name of normal fuzzy subsemigroups.Theorem 14.
LetG be a group with identity e, t∈[0,1] and let μ be a fuzzy subset in G. Then μ is a normal fuzzy subsemigroup with tip t of G if and only if μ is a normal fuzzy subgroup of G with tip t.Proof.
Observe thate is the unique idempotent in G and the inverse a-1 of a is certainly the unique weak inverse of a for all a∈G. If μ is a normal fuzzy subsemigroup with tip t of G, then by Proposition 13, μ is a normal fuzzy subgroup of G with tip t. Conversely, let μ be a normal fuzzy subgroup of G with tip t and a,x,y∈S. Then
(20)μ(a)=μ(x-1xayy-1)=μ(xayy-1x-1)≥μ(xay)∧μ(y-1x-1)
by condition (17). This implies that
(21)μ(a)≥μ(xay)∧μ(y-1x-1)=μ(xay)∧μ((xy)-1)=μ(xay)∧μ(xy).
Moreover, we have
(22)t=μ(e)=μ(aa-1)≥μ(a)∧μ(a-1)=μ(a)∧μ(a)=μ(a).
Thus, μ is a normal fuzzy subsemigroup of G with tip t.Since⋂i∈I‍σi∈NFSt(S) for σi∈NFSt(S), i∈I, and the element
(23)S⟶[0,1],a⟼t
is the greatest one in NFSt(S), we have the following theorem by Lemma 2.Theorem 15.
(
N
F
S
t
(
S
)
,
⊆
) forms a lattice.On normal fuzzy subsemigroups with tipt of E-inverse semigroups, we also have the following basic properties which will be used in the final section.Proposition 16.
Letμ∈NFSt(S), e,f∈E(S), a,b∈S, and a′,a*∈W(a),b′∈W(b).(1)
μ(ef)=t.(2)
μ(a′b)=μ(b′a).(3)
μ(a′b)=μ(a*b).Proof.
(
1
) Since e,f∈E(S), we have μ(e)=μ(f)=t. This implies that μ(ef)≥μ(e)∧μ(f)=t whence μ(ef)=t.(2)
The result follows from the facts that(24)μ(a′b)≥μ(aa′bb′)∧μ(ab′)=t∧μ(ab′)=μ(ab′)=μ(b′a),μ(b′a)=μ(ab′)≥μ(a′ab′b)∧μ(a′b)=t∧μ(a′b)=μ(a′b).(3)
This follows from the proof of Theorem 2.12 in Zhang [11].


## 5. The Relationship ofGFCt(S) and NFSt(S)

In this section, we show thatGFCt(S) is isomorphic to NFSt(S) as lattices whence NFSt(S) is modular for all t in [0,1]. We first give some useful propositions.Proposition 17.
Letμ∈NFSt(S) and
(25)ρμ:S×S⟶[0,1],(a,b)⟼μ(a′b).
Then ρμ∈GFCt(S), where a′∈W(a).Proof.
In view of Proposition16(3), the above ρμ is well defined. Now, let a,b,c∈S and a′∈W(a), b′∈W(b), c′∈W(c). Then we have the following facts:(1)
Sincea′a∈E(S), ρμ(a,a)=μ(a′a)=t.(2)
By Proposition16(2), we have
(26)ρμ(a,b)=μ(a′b)=μ(b′a)=ρμ(b,a)≤t.
(
3
) Since b′b∈E(S), it follows that μ(b′b)=t and
(27)μ(a′b)≥μ(ca′bb′)∧μ(cb′)≥μ(ca′)∧μ(bb′)∧μ(cb′)=μ(ca′)∧μ(cb′)=μ(a′c)∧μ(b′c)=μ(a′c)∧μ(c′b)
by Proposition 13(3) and Proposition 16(2). This implies that
(28)ρμ(a,b)≥ρμ(a,c)∧ρμ(c,b).
(
4
) For any (ac)′∈W(ac), we have ac(ac)′∈E(S) and
(29)μ((ac)′bc)≥μ(ac(ac)′bcc′b′)∧μ(acc′b′)≥μ(ac(ac)′)∧μ(bcc′b′)∧μ(acc′b′)=t∧μ(cc′b′b)∧μ(cc′b′a)=μ(cc′b′a)≥μ(cc′)∧μ(b′a)=t∧μ(b′a)=μ(b′a)=μ(a′b)
by Proposition 13(3) and Proposition 16(1), (2). This implies that μ(a′b)≤μ((ac)′bc). Dually, we have μ(a′b)≤μ((ca)′cb). This implies that
(30)μ(a′b)≤μ((ac)′bc)∧μ((ca)′cb).
Thus, ρμ(a,b)≤ρμ(ac,bc)∧ρμ(ca,cb).(5)
By Proposition16 and the fact that e∈W(e), we have ρμ(e,f)=μ(ef)=t.(6)
ρμ(a,aa′a)=μ(a′aa′a)=μ(aa′)=t.From the above six items, we can see that ρμ∈GFCt(S) by Proposition 6.Proposition 18.
Letρ∈GFCt(S) and define
(31)μρ:S⟶[0,1]:a⟼ρ(aa′,a).
Then μρ∈NFSt(S), where a′∈W(a).Proof.
By Proposition10(1), μρ is well defined. Now, let
(32)x,a,y∈S,x′∈W(x),y′∈W(y),(xy)′∈W(xy),(xay)′∈W(xay).
Then
(33)x′x,y′y,xx′,yy′,(xy)(xy)′∈E(S).
By Proposition 7, we have
(34)ρ((xy)(xy)′,xy)=ρ(yy′(xy)(xy)′,xy)=ρ(xx′yy′(xy)(xy)′,xy)≥ρ(xx′,x)∧ρ(yy′(xy)(xy)′,y)=ρ(xx′,x)∧ρ(yy′,y).
This implies that μρ(xy)≥μρ(x)∧μρ(y). On the other hand, also by Proposition 7, we have
(35)ρ(xay(xay)′,xay)∧ρ(xy(xy)′,xy)=ρ(xay,xay(xay)′xy(xy)′)∧ρ(xay(xay)′xy(xy)′,xy)≤ρ(xay,xy).
Observe that
(36)ρ(xay,xy)≤ρ(x′xayy′,x′xyy′)=ρ(x′xayy′,x′xyy′aa′)=ρ(a,aa′)=ρ(aa′,a)
by Proposition 7. Thus, t≥ρ(aa′,a)=μρ(a)≥μρ(xay)∧μρ(xy). Finally, since e∈W(e) for all e∈E(S), we have μρ(e)=ρ(ee,e)=ρ(e,e)=t for all e∈E(S). Therefore, μρ∈NFSt(S).At this stage, we can give the main result of this paper.Theorem 19.
The mappings(37)Ψt:GFCt(S)⟶NFSt(S),ρ⟼μρ;Φt:NFSt(S)⟶GFCt(S),μ⟼ρμ
are mutually inverse bijections preserving the inclusion relations, where μρ and ρμ are defined as in Propositions 17 and 18, respectively.Proof.
From Propositions17 and 18, the above mappings are well defined. Now, let μ∈NFSt(S), a∈S, and a′∈W(a). Then aa′∈W(aa′). This implies that
(38)ΨtΦt(μ)(a)=Ψt(ρμ)(a)=ρμ(aa′,a)=μ(aa′a)=μ(a)
by Proposition 13(2). On the other hand, for any a,b∈S and a′∈W(a),(a′b)′∈W(a′b), we have
(39)ΦtΨt(ρ)(a,b)=Φt(μρ)(a,b)=μρ(a′b)=ρ(a′b(a′b)′,a′b)=ρ(a,b)
by Proposition 10(2). This implies that Ψt and Φt are mutually inverse bijections. Obviously, Ψt and Φt preserve the inclusion relations.Corollary 20.
The lattices(GFCt(S),⊆) and (NFSt(S),⊆) are isomorphic. As a consequence, the lattice (NFSt(S),⊆) is also modular.Proof.
By Lemma1 and Theorem 19, the lattice (GFCt(S),⊆) is isomorphic to the lattice (NFSt(S),⊆). Thus, the lattice (NFSt(S),≤) is also modular by Theorem 9.Corollary 21.
(
G
F
C
(
S
)
,
⊆
) and (NFS(S),⊆) are two mutually isomorphic lattices.Proof.
It is routine to check that(40)(⋂i∈I‍ρi)∈GFC∧i∈Iti(S),(⋂j∈J‍σj)∈NFS∧j∈Jtj(S)
for ρi∈GFCti(S), i∈I, and σj∈NFStj(S), j∈J, where I and J are index sets. Moreover,
(41)ρ1:S×S⟶[0,1],(a,b)⟼1,σ1:S⟶[0,1],x⟼1
are the greatest elements in (GFC(S),⊆) and (NFS(S),⊆), respectively. By Lemma 2, (GFC(S),⊆) and (NFS(S),⊆) are two lattices. Moreover, if we let
(42)Ψ:GFC(S)⟶NFS(S),ρ⟼Ψt(ρ),ρ∈GFCt(S),Φ:NFS(S)⟶GFC(S),ρ⟼Φt(σ),σ∈NFSt(S),
then by Theorem 19, Ψ and Φ are mutually inverse bijections which preserve the inclusion relations and thus (GFC(S),⊆) and (NFS(S),⊆) are isomorphic from Lemma 1.We end this section by giving an example to illustrate our previous results.Example 22.
LetS be a semigroup with the following multiplication table:
(43)Saefbaeaaeeaeeafaefabeabe
Then S is an E-inversive semigroup which is nonregular. Moreover,
(44)E(S)={e,f},W(a)={a},W(e)={e,f},W(f)={f},W(b)={a}.Fix an elementt in the interval [0,1]. For every s∈[0,1] with s≤t, define a fuzzy set μs in S as follows:
(45)μs:S→[0,1],e⟼t,f⟼t,a⟼s,b⟼s.
It is routine to check that μs is a normal fuzzy subsemigroup with tip t in S. Furthermore, in view of the fact that
(46)E(S)={e,f},W(a)={a},W(e)={e,f},W(f)={f},W(b)={a}
and Proposition 13, we have
(47)NFSt(S)={μs∣s≤t,s∈[0,1]}.
Let μs∈NFSt(S). By Proposition 17, we can define
(48)ρμs:S×S⟶[0,1],(x,y)⟼μ(x′y),x′∈W(x).
More precisely, ρμs satisfies that
(49)ρμs(a,a)=ρμs(e,e)=ρμs(f,f)=ρμs(b,b)=ρμs(a,b)=ρμs(b,a)=ρμs(e,f)=ρμs(f,e)=t,ρμs(a,e)=ρμs(e,a)=ρμs(f,a)=ρμs(a,f)=ρμs(e,b)=ρμs(b,e)=ρμs(b,f)=ρμs(f,b)=s.
Then ρμs∈GFCt(S). By Theorem 19, GFCt(S)={ρμs∣s≤t,s∈[0,1]}. By virtue of Corollary 20, (GFCt(S),⊆) and (NFSt(S),⊆) are isomorphically modular lattices.


## 6. Conclusion

In this paper, we have introduced and investigated the lattices of group fuzzy congruences and normal fuzzy subsemigroups onE-inversive semigroups. Our results generalize the corresponding results of groups and regular semigroups. From the results presented in the paper, the lattices of group fuzzy congruences and normal fuzzy subsemigroups on E-inversive semigroups can be regarded as a source of possibly new modular lattices. On the other hand, this paper also leaves some questions which can be considered as future works. For example, from Ajmal and Thomas [8], if S is a group, then (NFS(S),⊆) is a modular lattice. Thus, the following question would be interesting: is (NFS(S),⊆) also modular for an E-inversive semigroup S?



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*Source: 413564-2014-05-05.xml*