When do L-fuzzy ideals of a ring generate a distributive lattice?

Ninghua Gao; Qingguo Li; Zhaowen Li

doi:10.1515/math-2016-0047

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When do L-fuzzy ideals of a ring generate a distributive lattice?

Ninghua Gao , Qingguo Li und Zhaowen Li

Veröffentlicht/Copyright: 23. Juli 2016

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 14 Heft 1

Abstract

The notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator “⇝” between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that “the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra” is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

Keywords: Boolean ring; Complete Heyting algebra; L-fuzzy extended ideals; L-fuzzy ideals

MSC 2010: 28A60; 06B05

1 Introduction

In 1971, Rosenfeld [1] applied the concept of fuzzy sets to abstract algebra and introduced the notion of fuzzy subgroups, and now the references related to various fuzzy algebraic substructures have been increasing rapidly. For example, Kuroki [2] investigated the properties of fuzzy ideals for a semigroup. Liu [3] introduced the fuzzy subring, etc. Standing upon these achievements, many researchers explored the lattice theoretical properties of these structures, such as, modularity of the lattice of the fuzzy normal subgroups was established in a systematic and step wise manner in [4–10]. By supposing the value “t” at the additive identity of a given ring, the fact that the set of fuzzy ideals with sup property forms a sublattice of the lattice of fuzzy ideals was proved by Ajimal and Thomas [11]. Furthermore, Majumdar and Sultana [12] also investigated the lattice of fuzzy ideals of a ring, and found that it is distributive, however, Zhang and Meng [13] pointed out that this result is erroneous. Subsequently, using a different proof from those of earlier papers, the author in [14] also stated that the lattice of all fuzzy ideals of a ring is modular. Modularity of the lattice of L-ideals of a ring was proved in [15], where L is a completely distributive lattice. Several authors have carried out further studies in this area (see Ref. [3, 16–25]).

Based on the above work, the question “When do L-fuzzy ideals of a ring generate a distributive lattice?” draws our attention. In fact, there exists a nontrivial class of rings so as to all of whose fuzzy ideals form a distributive lattice. The main purpose of this paper is to show that the family of all Boolean rings is such a class. For the Boolean ring, Gao and Cai [26] pointed out that it played a significant role in automata theory, which is the fundamental theory for the computer science technology.

The rest of this paper is organized as follows. In Section 2, we recall some fundamental notions and results to be used in the present paper. In Section 3, the definition of L-fuzzy extended ideals in a Boolean ring is introduced and basic properties are examined. In Section 4, we discuss the lattice structure of L-fuzzy ideals (LI(R)) in a Boolean ring by means of L-fuzzy extended ideals, moreover, the lattice structures of its three subsets are investigated. Conclusions are given in Section 5.

2 Preliminaries

We begin by recalling some definitions and results.

In a complete lattice L, for any S ⊆ L, write ⋁S for the lest upper bound of S and ⋀S the greatest lower bound of S.

Definition 2.1

Definition 2.1 ([27])

A residuated lattice is a structure

(L,∨,∧,⊗,→,0,1),

which satisfies the following conditions:

(L, ∨, ∧, 0, 1) is a bounded lattice with the least element 0 and the greatest element 1;
(L, ⊗, 1) is a commutative monoid with the identity 1;
(⊗, →) forms an adjoint pair, i.e., a ⊗ b ≤ c ⇔ a ≤ b → cfor anya, b ∈ L.

It is easy to check thata→b=⋁c∈L:a⊗c≤bfor anya,b ∈ L (see[28]). For example [0, 1] is a residuated lattice, in which for anyx, y ∈ [0, 1]:

x→y=⋁z∈L:z∧x≤y=1,x≤y,y,otherwise.

A residuated latticeLis called complete residuated if (L, ∧, ∨, 0, 1) is a complete lattice.

Throughout this paper, we denote by L a complete residuated lattice and R a Boolean ring unless otherwise stated. A ring R is called a Boolean ring if every element is idempotent (i.e., aa = a for all a ∈ R), such a ring is necessarily commutative and addition is modulo 2 (i.e., a + a = 0 for all a ∈ R, see [29]).

Properties of (complete) residuated lattices can be found in many papers, e.g. [27, 30–33]. We only give some which are used in the further text.

Proposition 2.2

Proposition 2.2 ([27, 32, 33])

In any complete residuated lattice (L, ∧, ∨, ⊗, 0,1), the following properties hold for anya, b, a_i, b_i, c ∈ L (i ∈ τ):

⊗ is isotone in both arguments, → antitone in the 1st argument and isotone in the 2nd argument;
a ≤ b → (a ⊗ b);
a ⊗ b ≤ a ∧ b;
a⊗⋁i∈τbi=⋁i∈τa⊗bi,a→⋀i∈τbi=⋀i∈τa→bi⋁i∈τai→b=⋀i∈τai→b;
(a ⊗ b) → c = b → (a → c) = a → (b → c);
a ≤ b ⇔ a → b = 1, 1 → a = a;
a ⊗ (b → c) ≤ (a → b) → c, specially, a ⊗ (a → b) ≤ b;
⋁i∈τai→bi≤(⋀i∈τai)→(⋁i∈τbi).

It is well known that a complete Heyting algebra (or frame) is a complete lattice L satisfying the following infinite distributive law:

a∧⋁B=⋁{a∧b|b∈B},∀a∈L,B⊆L.

Thus, a complete residuated lattice is a complete Heyting algebra (i.e., frame) if ⊗ = ∧.

An L-fuzzy subset of X is a function from X into L. The set of all L-fuzzy subsets of X is denoted by L^X. For any f, g ∈ L^X, f is called contained in g if f(x) ≤ g(x) for every x ∈ X, which is denoted by f ⊆ g. In particular, when L is [0,1], the L-fuzzy subsets of X are called fuzzy subsets of X.

Specially, for any A ⊆ X, the characteristic function χ_A is defined as follows:

χAy=1,y∈A,0,y∉A.

We denote by χ_x instead of χ{x}. Furthermore, 0¯,1¯∈LX are defined as follows:

0¯:X⟶Lbyx⟼0¯x:=0,1¯:X⟶Lbyx⟼1¯x:=1.

In [34], the author introduced the concept of L-fuzzy ideals of a commutative ring and gave some propositions (Propositions 2.4–2.6), where L is a completely distributive lattice. Here, we give the similar definition and results when L is a complete residuated lattice, which is a more general structure of truth values than a completely distributive lattice, and we omit the proof.

Definition 2.3

Let μ ∈ L^R. Then μ is called an L-fuzzy ideal of R if and only if for any x, y ∈ R, it satisfies the following conditions:

μ(x) ∧ μ(y) ≤ μ(x − y);
μ(x) ∧ μ(y) ≤ μ(xy);
μ(x) ≤ i(xy).

Denote LI(R) = {μ\μ is an L-fuzzy ideal of R}, obviously 0¯,1¯∈LI(R).

Proposition 2.4

For any family {μ_i}_i∈τ ⊆ LI(R), the intersection ⋂i∈τ is an L-fuzzy ideal of R.

Proposition 2.5

Let ν ∈ L^R. Then the L-fuzzy ideal generated by ν is defined to be the least L-fuzzy ideal of R which contains ν. It is denoted by 〈ν〉, that is

v=⋂v⊆μi{μi|μi∈LI(R)}.

Theorem 2.6

The set of all L-fuzzy ideals LI(R) is a complete lattice under the ordering of L-fuzzy subset inclusion, where for any {μ_i}_i∈τ ⊆ LI(R), the infimum and the supremum are defined as:

⋀i∈τμi=⋂i∈τμi,⋁i∈τμi=⋃i∈τμi.

3 L-fuzzy extended ideals

Let μ be an L-fuzzy ideal of R and ν ∈ L^R. We define the L-fuzzy extended ideal of μ associated with ν as follows:

∀x∈Rεμvx=⋀b∈Rvb→μxb.

Specially

∀x,y∈!Rεμχyx=⋀b∈Rχyb→μxb=μxy.

For any L-fuzzy ideal μ, L-fuzzy subset ν of R, we put = εμ = {ε_μ(ν)|ν ∈ L^R} and ε^ν = {ε_ν(ν) ∈ LI(R)}, respectively.

Proposition 3.1

Let μ be an L-fuzzy ideal and ν ∈ L^R. Then we have

ε_μ(ν) is an L-fuzzy ideal of R;
μ ⊆ ε_μ(ν).

Proof

(1) For any x, y ∈ R, μ is an L-fuzzy ideal of R, which implies

εμvx−y=⋀b∈Rvb→μx−yb=⋀b∈Rvb→μxb−yb≥⋀b∈Rvb→μxb∧μyb=εμvx∧εμv(y),
εμ(v)(xy)=⋀b∈Rv(b)→μ(xyb)=⋀b∈Rv(b)→μ(xybb)≥⋀b∈Rv(b)→(μ(xb)∧μ(yb))=εμ(v)x∧εμ(v)(y),
εμ(v)(xy)=⋀b∈Rv(b)→μ(xyb)≥⋀b∈Rv(b)→μ(xb)=εμ(v)x,

thus, ε_μ(ν) is an L-fuzzy ideal of R.

(2) For any x ∈ R,

εμ(v)(x)=⋀b∈Rv(b)→μ(xb)≥⋀b∈R1→μ(xb)=⋀b∈Rμ(xb)=μ(x)

by (I3), i.e., μ ⊆ε_μ (ν). □

Example 3.2

Let R = {0, p, q, r} with the following Cayley tables:

One can easily verify that it is a Boolean ring. Let L = {0, a, b, c, d, 1} be a complete lattice depicted in Figure 1.

$Fig. 1 The lattice L$

Fig. 1

The lattice L

The precomplement operator ¬ is given in Table 1.

Table 1

x	0	a	b	c	d	1
¬x	1	b	a	d	c	0

The generalized triangular norm ⊗ and the implication operator → in L are defined as follows: for any x, y ∈ L,

x⊗y=0,x≤¬y,x∧y,x¬y.x→y=1,x≤y,¬x∨y,xy.

Then (L, ∨, ∧, ⊗, →, 0, 1) is a complete residuated lattice.

We check that an L-fuzzy subset

f=t10,t2p,t3q,t4r(ti∈L,i=1,2,3,4)

of R is an L-fuzzy ideal of R if and only if t₁ ≥ t₂, t₃, t₄ and t₄= t₂ ∧ t₃. Now, considering the L-fuzzy ideal

f0=d0,ap,bq,cr

and the L-fuzzy subset

v=00,dp,aq,br,

we can calculate

εfov=10,ap,bq,cr.

Definition 3.3

An L-fuzzy ideal μ is called stable relative to the L-fuzzy subset ν if μ = ε_μ(ν).

For any L-fuzzy subset ν, denote S(ν) = {μ ∈ LI(R) | ε_μ(ν) = μ}.

Example 3.4

Let R and L be the Boolean ring and complete residuated lattice defined as in Example 3.1, respectively. For the L-fuzzy ideal

f1=10,ap,bq,cr

and the L-fuzzy subset ν in Example 3.1, we have

εf1v=10,ap,bq,cr=f1.

i.e., f₁ is an L-fuzzy ideal stable relative to ν.

Next we present basic properties of L-fuzzy extended ideals in a Boolean ring.

Proposition 3.5

Let μ, μ₁, μ₂ be L-fuzzy ideals and μ, μ₁,ν₂, ω L-fuzzy subsets of R. We have

εμ0¯=ε1¯v=1¯. Moreover, if R has identity e, then ε_μ(χ_e) = μ;
if ν₁ ⊆ ν₂ then ε_μ(ν₂) ⊆ ε_μ(ν₁);
if μ₁ ⊆ μ₂ then ε_μ2(ν);
ν ⊆ ε_μ(ε_μ(ν));
εμ0¯=ε1¯v=1¯
ε_μ(ν) = ε_μ(ε_μ(ν)));
if ν ⊆ μ, thenεμv=1¯Furthermore, if R has identity e, thenεμv=1¯if and only if ν ⊆ μ;
if L is a complete Heyting algebra and μ₁ ⊆ μ₂, then ε_μ1 (μ₂) ⋂ μ₂ = μ₁;
1. εμ(⋃i∈τνi)=⋂i∈τεμ(νi),,
2. ⋂i∈τεμi(ν)=ε∩i∈τμi(ν)
if L is a complete Heyting algebra, thenεεμ(ν)ν=εμν;
(relative extension property) let ν₁ ⊆ ν₂, then if μ is an L-fuzzy stable ideal relative to ν₁, μ is an L-fuzzy stable ideal relative to ν₂, too.

Proof

It can be easily obtained by the definition of L-fuzzy extended ideals.
Assume that ν₁ ⊆ ν₂, then for any x ∈ R,
εμ(v2)(x)=⋀b∈Rv2(b)→μ(xb)≤⋀b∈Rv1(b)→μ(xb)=εμ(v1)(x),
i.e., ε_μ(ν₂) ⊆ ε_μ(ν₁).
Let μ₁ ⊆ μ₂. For any x ∈ R,
εμ1(v)(x)=⋀b∈Rv(b)→μ1(xb)≤⋀b∈Rv(b)→μ2(xb)=εμ2(v)(x),
i.e., μ₁ (ν) ⊆ ε_μ2(ν).
For any x ∈ R,
εμ(εμ(v))(x)=⋀b∈Rεμ(v)(b)→μ(xb)=⋀b∈R⋀c∈Rv(c)→μ(bc)→μ(xb)≥⋀b∈R(v(x)→μ(bx))→μ(xb)≥⋀b∈Rv(x)⊗(μ(bx)→μ(xb))=v(x),
hence, ν ⊆ ε_μ(ε_μ(ν)).
For any x ∈ R, we get
εεμ(v)(ω)(x)=⋀b∈Rw(b)→εμ(v)(xb)=⋀b∈Rw(b)→∧c∈R⁡(v(c)→μ(xbc))=⋀b∈R⋀c∈R(w(b)→(v(c)→μ(xbc)))=⋀b∈R⋀c∈R(v(c)→(w(b)→μ(xbc)))=⋀c∈Rv(c)→∧b∈R⁡(w(b)→μ(xbc))=⋀c∈Rv(c)→εμ(ω)(xc)=εεμ(ω)(v)(x),
therefore, εεμ(ν)ω=εεμ(ω)ν.
It follows from (2) and (4).
For any x ∈ R, by (I3), we have
εμ(v)(x)=⋀b∈Rv(b)→μ(xb)≥⋀b∈Rv(b)→μ(b)=1.
If R has identity e, assuming εμν=1¯, i.e., for any x ∈ R, ν(b) ≤ μ(xb) for all b ∈ R, pick x = e, we have ν⊆ μ
Let μ₁ ⊆ μ₂. We only prove εμ1μ2∩μ2⊆μ1. L is a complete Heyting algebra, which implies that for any x ∈ R,
εμ1μ2∩μ2x=εμ1μ2x∧μ2x=⋀b∈Rμ2b→μ1xb∧μ2x≤μ2x→μ1xx∧μ2x=μ2x→μ1x∧μ2x=μ1x.
For any x ∈ R,
1. εμ(⋃i∈τvi)(x)=⋀b∈R⋁i∈τvi(b)→μ(xb)=⋀i∈τ⋀b∈Rvi(b)→μ(xb)=⋂i∈τεμ(vi),
  i.e., εμ(⋃i∈τνi)=⋂i∈τεμνi.
2. ⋂i∈τεμi(v)(x)=⋀i∈τεμi(v)(x)=⋀i∈τ⋀b∈Rv(b)→μi(xb)=⋀b∈Rv(b)→⋀i∈τμi(xb)=⋀b∈Rv(b)→(∩i∈τμi(xb))=ε∩i∈τμi(v),
  i.e., ⋂i∈τεμi(ν)=ε∩i∈τμi(ν).
It is sufficient to prove εεμ(ν)ν⊆εμν, for any x ∈ R,
εεμ(v)(v)(x)=⋀b∈R(v(b))→εμ(v)(x(b))=⋀b∈Rv(b)→⋀c∈Rv(c)→μ(xbc)≤⋀b∈R(v(b)→(v(b)→μ(xbb)))=⋀b∈R(v(b)→(v(b)→μ(xb)))=⋀b∈R((v(b)∧v(b))→μ(xb))=⋀b∈R(v(b)→μ(xb))=εμ(v),
i.e., εεμ(ν)ν=εμν.
This is a direct result of (2) in Proposition 3.1 and (2) of this proposition. □

The following example indicates that some equations corresponding to (9) in Proposition 3.2 are not always true, which contributes to studying the lattice structure in Section 4.

Example 3.6

Let R and L be the Boolean ring and complete residuated lattice defined as in Example 3.1, respectively.

For the L-fuzzy ideal f₀, L-fuzzy subsets ν in Example 3.1 and the L-fuzzy ideal
f2=a0,cq,aq,cr,
we obtain
f0∪f2=d0,ap,dq,cr,f0∪f2=d0,ap,dq,ar,εf2(v)=a0,cp,aq,cr,εf0∪f2(v)=10,ap,1q,ar≠εf0(v)∪εf2(v)=εf0(v)∪εf2(v)=10,ap,dq,cr,
i.e., εfoν∪εf2ν≠εf0∪f2ν.
Considering the L-fuzzy ideal f₀, L-fuzzy subset ν in Example 3.1 and the L-fuzzy subset
ω=10,bp,bq,ar,
we get
v∩ω=00,bp,cq,cr,εf0(ω)=d0,ap,dq,cr,εf0(v∩ω)=10,ap,1q,ar≠εf0(v)∪εf0(ω)=εf0(v)∪εf0(ω)=10,ap,dq,cr,
i.e., εfoν∪εf0ω≠εf0ν∩ω.

For any L-fuzzy ideal μ of R, we present the following characterization theorem via L-fuzzy extended ideals.

Theorem 3.7

Let μ be an L-fuzzy ideal of R. Then μ = ∩ ε_μ.

Proof

We need to prove μ = ∩ {ε_μ(ν)|ν ∈ L^R}. Obviously, μ⊆⋂ν∈LRεμ(ν) by (2) in Proposition 3.1. On the other hand, for any x ∈ R, we have

⋂v∈LRεμ(v)(x)=⋀v∈LRεμ(v)(x)≤εμ(χx)(x)=μ(xx)=μ(x),

thus, μ = ∩ ε_μ. □

4 Lattice structures

As mentioned in Theorem 2.1, the class of L-fuzzy ideals of a commutative ring is a complete lattice. In this section, three other subsets of this class are also investigated in a Boolean ring R. Moreover, in terms of L-fuzzy extended ideals, we show that all L-fuzzy ideals of R form a complete Heyting algebra, thus a distributive lattice. The following information are reviewed in order to discuss the lattice structures.

In a poset P, for any S ⊆ P, we denote S^u = {y |x ≤ y (∀x ∈ S)}.

Proposition 4.1

Proposition 4.1 ([35])

Let P be a poset such that⋀Sexists in P for every non-empty subset S of P. Then⋁Qexists for every non-empty subset Q of P, indeed, ⋁Q=⋀Qu.

Theorem 4.2

Theorem 4.2 ([35])

Let P be a non-empty ordered set. Then the following are equivalent:

P is a complete lattice;
⋀Sexists in P for every subset S of P;
P has a top element, and⋀Sexists in P for every non-empty subset S of P.

Definition 4.3

Definition 4.3 ([35])

Let P be an ordered set. A closure operator is a mapping c : P → P satisfying for every a, b ∈ P,

a ≤ c(a);
a ≤ b ⇒ c(a) ≤ c(b);
c(c(a)) = c(a).

Denote P_c = {x ∈ P | c(x) = x}.

Proposition 4.4

Proposition 4.4 ([35])

Let c be a closure operator on an ordered set P. Then

P_c = {c(x) | x ∈ P};
for anyx∈P,c(x)=⋀P⁡{y∈Pc|x≤y};
P_c is a complete lattice, under the order inherited from P, such that, for every subset S of P_c:
⋀PcS=⋀!PS,⋁PcS=c(⋁PS).

Theorem 4.5

Let μ be an L-fuzzy ideal of R and ν ∈ L^R. We have

if R has identity e, then (ε_μ, ⊆) is a complete lattice with the least element ε_μ(χ_e) and the greatest elementεμ(0¯). Moreover, for any {ε_μ(ν_i)}, _i∈τ ⊆ ε_μ:
⋀i∈τεμ(vi)=εμ(⋃i∈τvi),⋁i∈τεμ(vi)=⋂{εμ(vi)|i∈τ}u.
(ε^ν, ⊆) is a complete lattice with the least elementε0¯(ν)and the greatest elementε1¯(ν), and for any{εμi(ν)}i∈τ⊆εν:
⋀i∈τεμi(v)=ε∩i∈τμi(v),⋁i∈τεμi(v)=⋂{εμi(v)|i∈τ}u.
(S(ν), ⊆) is a sub-complete lattice of (ε^ν, ⊆) with the greatest elementε1¯(ν). Furthermore, for any {μ_i}_{i∈ τ} ⊆ S(ν),
⋀i∈!τμi=⋂i∈τμi,⋁i∈τμi=⋂i∈τ{μi|i∈τ}u.

Proof

It is trivial that 1¯=εμ(0¯)∈εμ. According to (9) (a) in Proposition 3.2, for any {εμ(νi)}i∈τ⊆εμ,⋂i∈τεμ(νi)=εμ(⋃i∈τνi)∈εμ. This completes the proof by (1) in Proposition 3.2, Theorem 4.1 and Proposition 4.1.
It is obvious that for any μ∈LI(R),ε0¯(ν)⊆εμ(ν) by (3) in Proposition 3.2, and 1¯=ε1¯(ν)∈εν. For any {ε_μi (ν)}_i∈τ ⊆ ε^ν, from (9) (b) in Proposition 3.2, it follows that ⋂i∈τεμi(ν)=ε∩i∈τμi(ν)∈εν. Similar to the proof of (1), the proof is completed.
Immediately, S(ν) ⊆ ε^ν and ε1¯(ν)∈S(ν). For any {μi}i∈τ⊆S(ν),⋂i∈τμi=⋂i∈τεμi(ν)=ε∩i∈τμi(ν) by Theorem 2.1 and (9) (b) in Proposition 3.2, i.e., ⋂i∈τμi∈S(ν). Analogously, the proof is completed. □

For an L-fuzzy subset ν of R, the following example indicates that (S(ν), ⊆) is not always a complete sublattice of (ε^ν, ⊆).

Example 4.6

Let R be the Boolean ring, L the complete residuated lattice and ν the L-fuzzy subset defined as in Example 3.1, respectively. Considering the L-fuzzy ideals

g0=b0,cp,bq,cr

andf₂, where f₂is defined as in Example 3.3, we have

εg0(v)=g0=b0,cp,bq,cr,εf2(v)=f2=a0,cp,aq,cr,

i.e., g₀, f₂ ∈ S(ν). Putϱ=g0⋁f2=⋂{g0,f2}u, we can calculate

ϱ=d0,cp,dq,cr,

however,

εϱ(v)=10,cp,1q,cr≠ϱ,i.e.,ϱ∉S(v).

Theorem 4.7

Let L be a complete Heyting algebra. Then(LI(R),∧,∨,⇝,0¯,1¯)is a complete Heyting algebra, thus a distributive lattice, where for any {μ_i}_i∈τ ⊆ LI(R), ∧, ∨ are defined as in Theorem 2.1 and for any μ, ϱ ∈ LI(R), ⇝ is defined as:

μ⇝ϱ=εϱ(μ).

Proof

It is sufficient to prove that for any μ, ϱ, σ ∈ LI(R), μ ∧ ϱ ⊆ σ ⇔ μ ⊆ ϱ ⇝ σ. (⇒) For any x ∈ R, we get

(ϱ⇝σ)(x)=εσ(ϱ)(x)=⋀b∈R(ϱ(b)→σ(xb))≥⋀b∈R(ϱ(b)→(μ∩ϱ)(xb))=⋀b∈R(ϱ(b)→(μ(xb)∧ϱ(xb)))≥⋀b∈R(ϱ(xb)→(μ(xb)∧ϱ(xb)))≥⋀b∈Rμ(xb)≥μ(x),

i.e., μ ⊆ ϱ ⇝ σ.

(⇐) L is a complete Heyting algebra, which implies that for any x ∈ R,

(μ∧ϱ)(x)=μ(x)∧ϱ(x)≤(ϱ⇝σ)(x)∧ϱ(x)=εσ(ϱ)(x)∧ϱ(x)=⋀b∈R(ϱ(b)→σ(xb))∧ϱ(x)≤(ϱ(x)→σ(xx))∧ϱ(x)=(ϱ(x)→σ(x))∧ϱ(x)≤σ(x),

i.e., μ ∧ ϱ ⊆ σ. □

Theorem 4.8

Let L be a complete Heyting algebra and ν ∈ L^R. Then

ε(ν) is a closure operator on LI(R);
S(ν) = ε^ν;
in the complete lattice (S(ν) (resp., ε^ν), ⊆), the least element isε0¯(ν)and for any {μ_i}_i∈τ⊆S(ν),
⋁i∈τμi=ε∪i∈τμi(v);
if ν is an L-fuzzy ideal, then (S(ν)(resp., ε^ν), ⊆) is a complete Heyting algebra.

Proof

It follows from (2) in Proposition 3.1 and (3), (10) in Proposition 3.2.
It is trivial from (1) in this theorem and (1) in Proposition 4.2.
Obviously, for any μ∈LI(R),ε0¯(ν)⊆εμ(ν)⊆1¯=ε1¯(ν) according to (2) in Proposition 3.1. By (1) in this theorem and (3) in Proposition 4.1, for any {μi}i∈τ⊆S(ν)(resp.,εν),⋁i∈τμi=ε∪i∈τμi(ν).
We only need to prove that for any μ, ϱ, ∈ S(ν), ⇝ ϱ ∈ S(ν).

For any μ, ϱ, ∈ S(ν), we get μ = ε_μ(ν) and ϱ = ε_ϱ(ν). Then μ ⇝ ϱ = μ ⇝ ε_ϱ(ν) = μ ⇝ (ν ⇝ ϱ) = (ν ∧ μ) ⇝ ϱ = ν ⇝ (μ ⇝ ϱ) = ε _μ⇝ϱ (ν) ∈ ε^ν = S(ν) by Theorem 4.3 and (2) in this theorem. □

Remark 4.9

Let L be a complete Heyting algebra and ν ∈ LI(R). Then (1) in Example 3.3 illustrates that (S(ν) (resp., ε^ν), ⊆) may not be a subalgebra of(LI(R),⊓,⊔,⇝,0¯,1¯).

5 Conclusions

By the aid of L-fuzzy extended ideals, which are firstly introduced in this work, we conclude that if R is a Boolean ring, then the lattice of all its L-fuzzy ideals (LI(R), ⊆) is distributive. We will consider whether the converse is affirmative or not in our further work. In this paper, we have also obtained some other results such as: the family of L-fuzzy extended ideals of an L-fuzzy ideal μ (ε_μ, ⊆) forms a complete lattice, all L-fuzzy extended ideals relative to an L-fuzzy subset ν (ε^ν, ⊆) generate a complete lattice and the lattice of L-fuzzy stable ideals relative to an L-fuzzy subset ν (S(ν), ⊆) is a sub-complete lattice rather than a complete sublattice of (ε^ν, ⊆). In particular, if L is a complete Heyting algebra, then ε^ν and S(ν) coincide, and the class of L-fuzzy stable ideals relative to an L-fuzzy ideal produces a complete Heyting algebra, but it is not a subalgebra of (LI(R), ⊆).

Acknowledgement

This work is supported by the National Natural Science Foundation of China (No. 11371130, 11461005) and Research Fund for the Doctoral Program of Higher Education of China (No. 20120161110017).

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Received: 2016-4-7

Accepted: 2016-6-28

Published Online: 2016-7-23

Published in Print: 2016-1-1

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