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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

Issues

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

Ninghua Gao / Qingguo Li / Zhaowen Li
  • College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi, China 530006
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-07-23 | DOI: https://doi.org/10.1515/math-2016-0047

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 [410]. 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, 1625]).

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 SL, write S for the lest upper bound of S and S the greatest lower bound of S.

([27])

A residuated lattice is a structure

(L,,,,,0,1),

which satisfies the following conditions:

  1. (L, ∨, ∧, 0, 1) is a bounded lattice with the least element 0 and the greatest element 1;

  2. (L, ⊗, 1) is a commutative monoid with the identity 1;

  3. (⊗, →) forms an adjoint pair, i.e., abcabc for any a, bL.

It is easy to check that ab=cL:acb for any a,bL (see [28]). For example [0, 1] is a residuated lattice, in which for any x, y ∈ [0, 1]:

xy=zL:zxy=1,xy,y,otherwise.

A residuated lattice L is 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 aR), such a ring is necessarily commutative and addition is modulo 2 (i.e., a + a = 0 for all aR, see [29]).

Properties of (complete) residuated lattices can be found in many papers, e.g. [27, 3033]. We only give some which are used in the further text.

([27, 32, 33])

In any complete residuated lattice (L, ∧, ∨, ⊗, 0,1), the following properties hold for any a, b, ai, bi, cL (iτ):

  1. is isotone in both arguments, → antitone in the 1st argument and isotone in the 2nd argument;

  2. ab → (ab);

  3. abab;

  4. aiτbi=iτabi,aiτbi=iτabiiτaib=iτaib;

  5. (ab) → c = b → (ac) = a → (bc);

  6. abab = 1, 1 → a = a;

  7. a ⊗ (bc) ≤ (ab) → c, specially, a ⊗ (ab) ≤ b;

  8. iτaibi(iτai)(iτbi).

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

aB={ab|bB},aL,BL.

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 LX. For any f, gLX, f is called contained in g if f(x) ≤ g(x) for every xX, which is denoted by fg. In particular, when L is [0,1], the L-fuzzy subsets of X are called fuzzy subsets of X.

Specially, for any AX, the characteristic function χA is defined as follows:

χAy=1,yA,0,yA.

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

0¯:XLbyx0¯x:=0,1¯:XLbyx1¯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.

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

  • (I1)

    μ(x) ∧ μ(y) ≤ μ(xy);

  • (I2)

    μ(x) ∧ μ(y) ≤ μ(xy);

  • (I3)

    μ(x) ≤ i(xy).

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

For any family {μi}iτLI(R), the intersection iτ is an L-fuzzy ideal of R.

Let νLR. 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=vμi{μi|μiLI(R)}.

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

iτμi=iτμi,iτμi=iτμi.

3 L-fuzzy extended ideals

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

xRεμvx=bRvbμxb.

Specially

x,y∈!Rεμχyx=bRχybμxb=μxy.

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

Let μ be an L-fuzzy ideal and νLR. Then we have

  1. εμ(ν) is an L-fuzzy ideal of R;

  2. μ ⊆ εμ).

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.

The lattice L
Fig. 1

The lattice L

The precomplement operator ¬ is given in Table 1.

Table 1

The generalized triangular normand the implication operator → in L are defined as follows: for any x, yL,

xy=0,x¬y,xy,x¬y.xy=1,xy,¬xy,xy.

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

We check that an L-fuzzy subset

f=t10,t2p,t3q,t4r(tiL,i=1,2,3,4)

of R is an L-fuzzy ideal of R if and only if t1t2, t3, t4 and t4= t2t3. 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.

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

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

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.

Let μ, μ1, μ2 be L-fuzzy ideals and μ, μ12, ω L-fuzzy subsets of R. We have

  1. εμ0¯=ε1¯v=1¯. Moreover, if R has identity e, then εμ(χe) = μ;

  2. if ν1ν2 then εμ(ν2) ⊆ εμ(ν1);

  3. if μ1μ2 then εμ2(ν);

  4. νεμ(εμ(ν));

  5. εμ0¯=ε1¯v=1¯

  6. εμ) = εμ(εμ(ν)));

  7. if ν ⊆ μ, then εμv=1¯ Furthermore, if R has identity e, then εμv=1¯ if and only if ν ⊆ μ;

  8. if L is a complete Heyting algebra and μ1 ⊆ μ2, then εμ12) ⋂ μ2 = μ1;

    1. εμ(iτνi)=iτεμ(νi),,

    2. iτεμi(ν)=εiτμi(ν)

  9. if L is a complete Heyting algebra, then εεμ(ν)ν=εμν;

  10. (relative extension property) let ν1ν2, then if μ is an L-fuzzy stable ideal relative to ν1, μ is an L-fuzzy stable ideal relative to ν2, too.

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.

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

  1. 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

    f0f2=d0,ap,dq,cr,f0f2=d0,ap,dq,ar,εf2(v)=a0,cp,aq,cr,εf0f2(v)=10,ap,1q,arεf0(v)εf2(v)=εf0(v)εf2(v)=10,ap,dq,cr,

    i.e., εfoνεf2νεf0f2ν.

  2. 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

    vω=00,bp,cq,cr,εf0(ω)=d0,ap,dq,cr,εf0(vω)=10,ap,1q,arεf0(v)εf0(ω)=εf0(v)εf0(ω)=10,ap,dq,cr,

    i.e., εfoνεf0ωεf0νω.

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

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

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 SP, we denote Su = {y |xy (∀xS)}.

([35])

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

([35])

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

  1. P is a complete lattice;

  2. S exists in P for every subset S of P;

  3. P has a top element, and S exists in P for every non-empty subset S of P.

([35])

Let P be an ordered set. A closure operator is a mapping c : PP satisfying for every a, bP,

  1. ac(a);

  2. abc(a) ≤ c(b);

  3. c(c(a)) = c(a).

Denote Pc = {xP | c(x) = x}.

([35])

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

  1. Pc = {c(x) | xP};

  2. for any xP,c(x)=P{yPc|xy};

  3. Pc is a complete lattice, under the order inherited from P, such that, for every subset S of Pc:

    PcS=⋀!PS,PcS=c(PS).

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

  1. if R has identity e, thenμ, ⊆) is a complete lattice with the least element εμe) and the greatest element εμ(0¯). Moreover, for anyμi)}, iτ ⊆ εμ:

    iτεμ(vi)=εμ(iτvi),iτεμ(vi)={εμ(vi)|iτ}u.

  2. ν, ⊆) is a complete lattice with the least element ε0¯(ν) and the greatest element ε1¯(ν), and for any {εμi(ν)}iτεν:

    iτεμi(v)=εiτμi(v),iτεμi(v)={εμi(v)|iτ}u.

  3. (S(ν), ⊆) is a sub-complete lattice ofν, ⊆) with the greatest element ε1¯(ν). Furthermore, for anyi}i∈ τS(ν),

    i∈!τμi=iτμi,iτμi=iτ{μi|iτ}u.

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

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

and f2, where f2 is 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, f2S(ν). Put ϱ=g0f2={g0,f2}u, we can calculate

ϱ=d0,cp,dq,cr,

however,

εϱ(v)=10,cp,1q,crϱ,i.e.,ϱS(v).

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

μϱ=εϱ(μ).

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

  1. ε(ν) is a closure operator on LI(R);

  2. S(ν) = εν;

  3. in the complete lattice (S(ν) (resp., εν), ⊆), the least element is ε0¯(ν) and for anyi}iτS(ν),

    iτμi=εiτμi(v);

  4. if ν is an L-fuzzy ideal, then (S(ν)(resp., εν), ⊆) is a complete Heyting algebra.

Let L be a complete Heyting algebra and ν ∈ LI(R). Then (1) in Example 3.3 illustrates that (S(ν) (resp., εν), ⊆) 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 (εμ, ⊆) forms a complete lattice, all L-fuzzy extended ideals relative to an L-fuzzy subset ν (εν, ⊆) 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 (εν, ⊆). In particular, if L is a complete Heyting algebra, then εν 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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About the article

Received: 2016-04-07

Accepted: 2016-06-28

Published Online: 2016-07-23

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0047.

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© 2016 Long et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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