Characterizations of fuzzy ideals in coresiduated lattices.

1. Introduction

Residuated lattices provide an algebra frame for the algebraic semantics of formal fuzzy logics such as MV-algebras, BL-algebras, and [R.sub.0]-algebras (or MTL-algebras) [1-6]. Filters play crucial important role in the proof of the completeness of these logics. Many results about the filter theory of various fuzzy logical algebras have sprang out [7-11]. In fact, ideal and filter are considered as the dual notions of the logical algebra systems in the view of algebra structure. Ideal is also interesting because it is closely related to congruence relation. It plays a vital role in Chang's Subdirect Representation Theorem of MV-algebras . Actually, the original definition of MV-algebra was characterized by the operator [R] which could be looked as the generation of conorm in lattices. Obviously, there existed an operator [??] such that ([direct sum], [??]) constituted a pair just like adjoint pair ([direct sum], [right arrow] ). Therefore, the definition of coadjoint pair was presented in , and the definition of coresiduated lattice, as the dual algebra structure of residuated lattice, was introduced in . Coresiduated lattices provide a new frame for logic algebras. It was masterly used to obtain the unified form of intuitionistic fuzzy implications and the Triple I method solutions of intuitionistic fuzzy reasoning problems in . It was also used to obtain the unified form of intuitionistic fuzzy difference operators and the Triple D method solutions of intuitionistic fuzzy reasoning problems in . The properties of ideals in coresiduated lattices were investigated and the embedding theorem of coresiduated lattices was obtained in .

In recent years, fuzzy filters in various logic algebras have captured many scholars' attention. Liu and Li proposed the notion of fuzzy filters to BL-algebras and obtained their relative properties [16,17]. Jun et al. investigated fuzzy filters in MTL-algebras and lattice implication algebras . Zhu and Xu extended the fuzzy filters in BL-algebras and MTL-algebras to general residuated lattices .

In this paper, we intend to introduce the notions of fuzzy ideals in coresiduated lattices and develop the ideal theory of coresiduated lattices. The rest of the paper is structured as follows. In Section 2 we recall the basic notions and existing results that will be used in the paper. In Section 3 we introduce the concept of fuzzy ideals in coresiduated lattices and present the characterizations of fuzzy ideals. In Section 4 we introduce the concepts of fuzzy prime ideals and fuzzy strong prime ideals in coresiduated lattices and obtain some of their characterizations. In Section 5 we advance the fuzzy prime ideal and fuzzy strong prime ideal in prelinear coresiduated lattices.

2. Preliminaries

In this section, we recall some basic concepts and results, which we will need in the subsequent sections.

Definition 1 (see ). Let P be a poset. ([direct sum], [??]) is called a coadjoint pair on P if the following conditions are satisfied.

(1) [direct sum] : P x P [right arrow] P is isotone.

(2) [??]: P x P [right arrow] P is isotone on first variable and antitone on second variable.

(3) a [less than or equal to] b [direct sum] c iff a [??] c [less than or equal to] b, a, b, c [member of] P.

Definition 2 (see ). A structure (L; [disjunction], [conjunction], [direct sum], [??], 0, 1) is called a coresiduated lattice if the following conditions are satisfied.

(1) (L; [disjunction], [conjunction]) is abounded lattice, 0 is the smallest element, and 1 is the greatest element of L, respectively.

(2) ([direct sum], [??]) is a coadjoint pair on L.

(3) (L; [direct sum], 0) is a commutative monoid.

[direct sum] is called a t-conorm operator on L, and [??] is called a fuzzy difference operator on L. One denotes CRL as the set of all coresiduated lattices.

Theorem 3 (see ). Suppose (L; [disjunction], [conjunction], [direct sum], [??], 0,1) [member of] CRL; then

(1) a [??] 0 = a,

(2) a [less than or equal to] b iff a [??] b = 0,

(3) a [direct sum] b = 0 iff a = b = 0,

(4) (a [??] b) [??] c = (a [??] c) [??] b = a [??] (b [direct sum] c),

(5) (a [direct sum] b) [??] b [less than or equal to] a [less than or equal to] (a [??] b) [direct sum]b,

(6) (a [direct sum] b)[??] b [??] a = (a [direct sum] b) [??] a [??] b = 0,

(7) a [??] (b [conjunction] c) = (a [??] b) [disjunction] (a [??] c),

(8) (a [disjunction] b) [??] c = (a [??] c) [disjunction] (b [??] c),

(9) (a [disjunction] b) [??] b = a [??] b,

(10) a [??] (a [conjunction] b) = a [??] b,

(11) a [??] (a [??] b) [less than or equal to] a [conjunction] b,

(12) (a [??] b) [direct sum] b [greater than or equal to] a [disjunction] b,

(13) (a [direct sum] c) [??] (c [direct sum] b) [less than or equal to] a [??] b [less than or equal to] (a [??] c) [direct sum] (c [??] b).

Definition 4 (see ). Suppose L [member of] CRL, I [subset or equal to] L; I is an ideal of L if one has the following:

(1) 0 [member of] I.

(2) If x [member of] I, y [member of] L, and y [less than or equal to] x, then y [member of] I.

(3) If x [member of] I, y [member of] l, then x [direct sum] y [member of] I.

I is a real ideal of L if I [not equal to] L.A real ideal I of L is a prime ideal if

(4) [for all]x, y [member of] L, if x [conjunction] y [member of] I, then x [member of] I or y [member of] I.

A real ideal I of L is a strong prime ideal if

(5) [for all]x, y [member of] L, x [??] y [member of] I or y [??] x [member of] I.

Theorem 5 (see ). Suppose L [member of] CRL, I [subset or equal to] L; I is an ideal of L if and only if one has the following:

(1) 0 [member of] I.

(2) If x [member of] I, y [??] x [member of] I, then y [member of] I.

Theorem 6 (see ). Suppose L [member of] CRL. I is a real ideal of L and A is a (strong) prime ideal of L; if A [subset or equal to] I, then I is a (strong) prime ideal of L.

Proposition 7 (see ). Consider L [member of] CRL; if I is a strong prime ideal of L, then I is a prime ideal of L.

3. Fuzzy Ideals of Coresiduated Lattices

Definition 8. Suppose that f is a fuzzy set on L, L [member of] CRL. f is a fuzzy ideal of L if the following conditions are satisfied:

(P1) f(0) [less than or equal to] f(x), [for all]x [member of] L,

(P2) f(y) [less than or equal to] f(y[??]x) [disjunction] f(x).

Throughout the rest of this paper, unless otherwise stated, L always represents a coresiduated lattice and f always represents a fuzzy set on L.

Proposition 9. Consider L [member of] CRL and f is a fuzzy ideal of L; then f is isotone; that is,

(P3) x [less than or equal to] y [??] f(x) [less than or equal to] f(y).

Proof. [for all]x, y [member of] L, if x [less than or equal to] y, then x [??] y = 0. By (P1) and (P2), it follows that

f(x) [less than or equal to] f(x [??] y) [disjunction] f(y) = f(0) [disjunction] f(y) = f(y). (1)

Theorem 10. Consider L [member of] CRL and f is a fuzzy ideal of L if and only if

(P4) z [less than or equal to] x[direct sum]y [??] f(z) [less than or equal to] f(x) [disjunction] f(y).

Proof.

Sufficiency. Consider [for all]x, y [member of] L, 0 [less than or equal to] x [direct sum] x; it follows by (P4) that

f(0) [less than or equal to] f(x) [disjunction] f(x) = f(x). (2)

So (P1) holds. From Theorem 3(5), y [less than or equal to] (y [??] x) [direct sum]x; it follows by (P4) that

f(y) [less than or equal to] f(y [??] x) [disjunction] f(x) (3)

So (P2) holds.

Necessity. If z [less than or equal to] x [direct sum] y, then z [??] y [less than or equal to] x. Because f is a fuzzy ideal of L, then it follows from (P3) that f(z [??] y) [less than or equal to] f(x). According to (P2),

f(z) [less than or equal to] f(z[??]y) [disjunction] f(y). (4)

Thus f(z) [less than or equal to] f(x) [disjunction] f(y). So (P4) holds.

From Theorem 10, it is easy to get the following corollary.

Corollary 11. Consider L [member of] CRL and f is a fuzzy set on L; then the following conditions are equivalent:

(1) f is a fuzzy ideal of L; that is, (P1) and (P2) hold.

(2) x [less than or equal to] y [??] f(x) [less than or equal to] f(y); that is, (P3) holds.

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 12. Consider L [member of] CRL and f is a fuzzy ideal on L if and only if

(P5) z [less than or equal to] [x.sub.1] [direct sum] [x.sub.2] [direct sum]*** [direct sum][x.sub.n] [??] f(z) [less than or equal to] max[f([x.sub.1]), f([x.sub.2]),..., f([x.sub.n])}, n [greater than or equal to] 2.

Proof.

Sufficiency. If n = 2, it is obviously valid. If n > 2, let z = 0 and [x.sub.i] = x, 1 [less than or equal to] i [less than or equal to] n, [for all]x [member of] L, from 0 [less than or equal to] x [direct sum] x [direct sum] *** [direct sum] x; then f(0) [less than or equal to] f(x).

If z [less than or equal to] x [direct sum] y, let [x.sub.1] = x, [x.sub.2] = y, [x.sub.i] = 0, 2 < i [less than or equal to] n; then f(z) [less than or equal to] f(x) [disjunction] f(y) [disjunction] f(0) = f(x) [disjunction] f(y); that is, (P4) holds. Thus f is a fuzzy ideal on L.

Necessity. Consider n = 2; it is obviously valid. Suppose that it is valid if n = k, k [greater than or equal to] 2; that is,

Z [less than or equal to] [x.sub.1] [direct sum] [x.sub.2] [direct sum] *** [direct sum] [x.sub.k] [??] f(x) [less than or equal to] max {f{[x.sub.1]),f{[x.sub.2]), ..., f([x.sub.k])}. (5)

Then, put n = k + 1; if z [less than or equal to] [x.sub.1] [direct sum] [x.sub.2] [direct sum]*** [direct sum][x.sub.k] [direct sum] [x.sub.k+1], then

z [??] [x.sub.k+1] [less than or equal to] [x.sub.1] [direct sum] [x.sub.2] [direct sum]***[direct sum][x.sub.k]. (6)

So

f(z [??] [x.sub.k+1]) [less than or equal to] max{f ([x.sub.1]), f([x.sub.2]), ..., f ([x.sub.k])}. (7)

Since f is a fuzzy set on L, it is obtained that f(z) [less than or equal to] f(z [??] [x.sub.k+1]) [disjunction] f([x.sub.k]). Therefore,

f(z) [less than or equal to] max {f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.k+1])}. (8)

The proof is completed.

Theorem 13. Consider L [member of] CRL and f is a fuzzy ideal of L if and only if

(1) f is isotone; that is, (P3) holds;

(2) (P6)f(x[direct sum]y) = f(x) [disjunction] f(y).

Proof.

Sufficiency. Suppose that (P3) and (P6) hold; it should be proved that (P4) holds. If z [less than or equal to] x [direct sum] y, it follows by (P3) and (P6) that

f(z) [less than or equal to] f(x[direct sum]y) = f(x) [disjunction] f(y). (9)

So (P4) holds.

Necessity. If f is a fuzzy ideal of L, it follows from Proposition 9 that f is isotone. Since x [less than or equal to] x [direct sum] y, y [less than or equal to] x [direct sum] y, then f(x) [less than or equal to] f(x[direct sum]y), f(y) [less than or equal to] f(x[direct sum]y). Thus f(x) [disjunction] f(y) [less than or equal to] f(x[direct sum]y).

Moreover, x [direct sum] y [less than or equal to] x [direct sum] y; it follows from (P4) that f(x [direct sum] y) [less than or equal to] f(x) [disjunction] f(y). Therefore, f(x [direct sum] y) = f(x) [direct sum] f(y). The proof is completed.

Let f be a fuzzy set on the coresiduated lattice L; it is denoted that

[f.sub.[lambda]] = {x [member of] L|f(x)[less than or equal to][lambda]}, [lambda] [member of] [0,1]. (10)

Theorem 14. f is a fuzzy ideal of L if and only if [for all][lambda] [member of] [0,1], [f.sub.[lambda]] is either an empty set or an ideal of L.

Proof.

Sufficiency. [for all]x, y [member of] L, suppose that f(x) = [[lambda].sub.0], and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is obvious that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not an empty set; then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an ideal of L, so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus f(0) [less than or equal to] [[lambda].sub.0] = f(x); that is, (P1) holds. Put [[lambda].sub.1] = f(x) [disjunction] f(y [??] x); then f(x) [less than or equal to] [[lambda].sub.1], f(y [??] x) [less than or equal to] [[lambda].sub.1]; that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an ideal of L, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; that is, f(y) [less than or equal to] [[lambda].sub.1] = f(x) [disjunction] f(y [??] x); thus (P2) holds. Therefore, f is a fuzzy ideal of L.

Necessity. Suppose that [f.sub.[lambda]] is not an empty set; then [there exists][x.sub.0] [member of] [f.sub.[lambda]] s.t. f([x.sub.0]) [less than or equal to] [lambda]. Since f is a fuzzy ideal of L, then f(0) [less than or equal to] f([x.sub.0]) [less than or equal to] [lambda]; that is, 0 [member of] [f.sub.[lambda]]. Suppose x, y [??] x [member of] [f.sub.[lambda]]; then f(x) [less than or equal to] [lambda], f(y [??] x) [less than or equal to] [lambda]. Thus f(y) [less than or equal to] f(x) [disjunction] f(y [??] x) [less than or equal to] [lambda] that is, y [member of] [f.sub.[lambda]]. It follows from Theorem 5 that [f.sub.[lambda]] is an ideal of L.

Corollary 15. Suppose that f is a fuzzy ideal of L; then [for all]a [member of] L,

[[OMEGA].sub.a] = {x [member of] L|f(x) [less than or equal to] f(a)} (11)

is an ideal of L.

Proof. Since f(a) [less than or equal to] f(a), it is obtained that a [member of] [[OMEGA].sub.a]; that is, [[OMEGA].sub.a] [not equal to] 0. It follows from Theorem 14 that [[OMEGA].sub.a] is an ideal of L.

Corollary 16. Suppose that f is a fuzzy ideal of L; then

[[OMEGA].sub.f] = {x [member of] L|f(x) = f(0)} (12)

is an ideal of L.

Proof. Since f is a fuzzy ideal of L, then f(0) [less than or equal to] f(x), x [member of] L. So

[[OMEGA].sub.f] = {x [member of] L|f(x) = f(0)} = {x [member of] L|f(x) [less than or equal to] f(0)} = [[OMEGA].sub.0] (13)

By Corollary 15, [[OMEGA].sub.0] is an ideal of L. Thus is an ideal of L.

Theorem 17. f is a fuzzy set on L, a [member of] L. If f satisfies the conditions

(1) f is isotone,

(2) [for all]x, y [member of] L, f(x) [less than or equal to] f(a),f(y) [less than or equal to] f(a) [??] f(x[direct sum]y) [less than or equal to] f(a),

then [[OMEGA].sub.a] is an ideal of L.

Proof. Since 0 [less than or equal to] a and f is isotone, then f(0) [less than or equal to] f(a); that is, 0 [member of] [[OMEGA].sub.a]. If x [member of] [[OMEGA].sub.a] and y [less than or equal to] x, then f(y) [less than or equal to] f(x) [less than or equal to] f(a); that is, y [member of] [[OMEGA].sub.a]. According to (2), x, y [member of] [[OMEGA].sub.a] [??] x [direct sum] y [member of] [[OMEGA].sub.a]. Therefore, [[OMEGA].sub.a] is an ideal of L.

4. Fuzzy Prime Ideals and Fuzzy Strong Prime Ideals of Coresiduated Lattices

Definition 18. Suppose that f is a fuzzy ideal of L and not a constant. f is a fuzzy prime ideal of L if the following conditions are satisfied:

(P7) f(x [conjunction] y) = f(x) [conjunction] f(y), [for all]x, y [member of] L.

In this section, it is always assumed that f is a fuzzy set on L and not a constant.

Theorem 19. f is a fuzzy prime ideal of L if and only if one has the following: if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [f.sub.[lambda]] is a prime ideal of L.

Proof.

Sufficiency. Since f is not a constant, then [there exists][lambda] [member of] [0,1] s.t. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So [f.sub.[lambda]] is a prime ideal of L. If [f.sub.[lambda]] is a prime ideal of L, then it is obviously an ideal of L. From Theorem 14, f is a fuzzy ideal of L. So f is isotone. [for all]x, y [member of] L, put f(x [conjunction] y) = [[lambda].sub.0]; then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a prime ideal, it follows from Definition 4 that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; then f(x) [less than or equal to] [[lambda].sub.0] = f(x [conjunction] y) [less than or equal to] f(x) [conjunction] f(y). By f(x) [conjunction] f(y) [less than or equal to] f(x), f(x [conjunction] y) = f(x) [conjunction] f(y); that is, f is a fuzzy prime ideal of L.

Necessity. [for all][lambda] [member of] [0,1],if x [conjunction]y [member of] [f.sub.[lambda]], then f(x[conjunction]y) [less than or equal to] A. Because f is a fuzzy prime ideal of L, f(x [conjunction] y) = f(x)[conjunction]f(y) [less than or equal to] A. Thus f(x) [less than or equal to] A or f(y) [less than or equal to] [lambda]; that is, x [member of] [f.sub.[lambda]] or y [member of] [f.sub.[lambda]]. Therefore, [f.sub.[lambda]] is a prime ideal of L.

Definition 20. Suppose that f is a fuzzy ideal of L and not a constant. f is a fuzzy strong prime ideal of L if the following conditions are satisfied:

(P8) f(x [??] y) = f(0) or f(y [??] x) = f(0), [for all]x, y [member of] L.

Theorem 21. f is a fuzzy strong prime ideal of L if and only if one has the following: if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [f.sub.[lambda]] is a strong prime ideal of L.

Proof.

Sufficiency. Since f is not a constant, then [there exists][lambda] [member of] [0,1] s.t. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So [f.sub.[lambda]] is a strong prime ideal of L. If [f.sub.[lambda]] is a strong prime ideal of L, then it is obviously an ideal of L. From Theorem 14, f is a fuzzy ideal of L. So f is isotone. [for all]x, y [member of] L, put f(0) = [[lambda].sub.0]; then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a strong prime ideal, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; then f(x [??] y) [less than or equal to] [[lambda].sub.0] = f(0). Because f is isotone, f(0) [less than or equal to] f(x [??] y). It is obtained that f is a fuzzy strong prime ideal of L.

Necessity. Since f is a fuzzy strong prime ideal of L, it is obvious that f is a fuzzy ideal of L. According to Theorem 14, [for all][lambda] [member of] [0,1], if [f.sub.[lambda]] is an ideal of L, then 0 [member of] [f.sub.[lambda]]; that is, f(0) [less than or equal to] X. Moreover, f is a fuzzy strong prime ideal of L.

If f(x p??[ y) = f(0), then f(x [??] y) [less than or equal to] [lambda]; that is, x [??] y [member of] [f.sub.[lambda]]. Therefore, [f.sub.[lambda]] is a strong prime ideal of L.

Theorem 22. If f is a fuzzy strong prime ideal of L then f is a fuzzy prime ideal of L.

Proof. f is a fuzzy strong prime ideal of L. [for all]x, y [member of] L, let f(x [??] y) = f(0). It follows from Theorem 13,

f ((x [conjunction] y) [direct sum] (x [??] y)) = f(x [conjunction] y) [disjunction] f(x [??] y) = f(x [conjunction] y) [disjunction] f(0) = f(x [conjunction] y).

By Theorem 3(5) and (7),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Since f is isotone, then f((x [conjunction] y) [direct sum] (x [??] y)) [greater than or equal to] f(x) [greater than or equal to] f(x) [conjunction] f(y). Thus f(x [conjunction] y) [greater than or equal to] f(x) [conjunction] f(y). Obviously, f(x [conjunction] y) [less than or equal to] f(x) [conjunction] f(y). So f(x [conjunction] y) = f(x) [conjunction] f(y); that is, (P6) holds. Therefore f is a fuzzy prime ideal of L.

Suppose that I is a real ideal of L; put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

According to Theorem 14, [X.sub.I] is a fuzzy ideal and not a constant. By Theorems 19 and 21, we can get the following two theorems, respectively.

Theorem 23. I is a prime ideal of L if and only if [X.sub.I] is a fuzzy prime ideal of L.

Theorem 24. I is a strong prime ideal of L if and only if [X.sub.I] is a fuzzy strong prime ideal of L.

5. Fuzzy Ideals and Fuzzy Prime Ideals of Prelinear Coresiduated Lattices

In a residuated lattice L, the condition

(x [right arrow] y) [disjunction] (y [right arrow] x) = 1, x, y [member of] L (17)

is called prelinearity axiom (see ). Taking account of coresiduated lattice as the dual algebra structure of residuated lattice, the definition of prelinear coresiduated lattice is as follows.

Definition 25 (see ). A coresiduated lattice L is called a prelinear coresiduated lattice if, [for all]x, y [member of] L, (x [??] y)[conjunction](y [??] x) = 0.

Theorem 26 (see ). Suppose that L is a prelinear coresiduated lattice; I is an ideal of L and a [member of] L \ I; then there exists a prime ideal J of L such that I [subset or equal to] J and a [bar.[member of]] J.

Theorem 27. L is a prelinear coresiduated lattice. f is a fuzzy strong prime ideal of L if and only if f is a fuzzy prime ideal of L.

Proof. Necessity is obvious by Theorem 22.

Sufficiency. [for all]x, y [member of] L, since L is a prelinear coresiduated lattice, then (x [??] y) [conjunction] (y [??] x) = 0. Because f is a fuzzy prime ideal of L,

f (x [??] y) [conjunction] f(y [??] x) = f((x [??] y) [conjunction] (x [??] y)) = f(0). (18)

Thus f(x [??] y) = f(0) or f(y [??] x) = f(0).

Corollary 28. Suppose that L is a prelinear coresiduated lattice and f is a fuzzy ideal of L; then the following conditions are equivalent:

(1) f is a fuzzy prime ideal of L.

(2) f(x [conjunction] y) = f(0) [??] f(x) = 0 or f(y) = 0.

(3) f is a fuzzy strong prime ideal of L.

Proof. It is only proved that (1) [??] (2) and (2) [??] (3).

(1) [??] (2): Since f is a fuzzy prime ideal of L, then f(x [conjunction] y) = f(x) [conjunction] f(y). Because f(x [conjunction] y) = f(0), f(x) [conjunction] f(y) = 0. Thus f(x) = 0 or f(y) = 0.

(2) [??] (3): L is a prelinear coresiduated lattice; then (x [??] y)[conjunction](y[??]x) = 0. So f((x [??] y) [conjunction] (x [??] y)) = f(0). According to (2), it follows that f(x [??] y) = f(0) or f(y [??] x) = f(0).

Remark 29. According to Theorem 27, we do not distinguish between fuzzy strong prime ideal and fuzzy prime ideal in prelinear coresiduated lattices.

Theorem 30. Suppose that L is a prelinear coresiduated lattice; f is a fuzzy ideal and not a constant; then the following conditions are equivalent:

(1) L is a chain.

(2) Any fuzzy ideal which is not a constant is a fuzzy prime ideal.

(3) Any fuzzy ideal such that f(0) = 0 and not a constant is a fuzzy prime ideal.

(4) The fuzzy ideal [X.sub.{0}] is a fuzzy prime ideal.

Proof. (1) [??] (2): [for all] x, y [member of] L, since L is a chain, then x [less than or equal to] y or y [less than or equal to] x. Thus x [??] y = 0 or y [??] x = 0. So f(x [??] y) = f(0) or f(y [??] x) = f(0). Therefore f is a fuzzy prime ideal.

It is obvious that (2) [??] (3) and (3) [??] (4). (4) [??] (1): [X.sub.{0}] is a fuzzy prime ideal; then [for all]x, y [member of] L, [X.sub.{0}](x [??] y) = [X.sub.{0}](0) = 0 or [X.sub.{0}](y [member of] x) = [X.sub.{0}](0) = 0. It is obtained that x [??] y [member of] {0} or y [??] x [member of] {0}; that is, x [??] y = 0 or y [??] x = 0. So x[less than or equal to]y or y [less than or equal to] x; that is, L is a chain.

Theorem 31. Suppose that L is a prelinear coresiduated lattice and f is a fuzzy prime ideal. If g is a fuzzy ideal and not a constant satisfying g [less than or equal to] f, g(0) = f(0), then g is a fuzzy prime ideal.

Proof. Since f is a fuzzy prime ideal, then, [for all]x, y [member of] L, f(x [??] y) = f(0) or f(y [??] x) = 0. Let f(x [??] y) = f(0). It follows from g [less than or equal to] f, g(0) = f(0) that

g(x [??] y) [less than or equal to] f(x [??] y)=f(0) = g(0). (19)

Because g is a fuzzy ideal, g is isotone. Thus g(0) [less than or equal to] g(x [??] y). So g(x [??] y) = g(0). Therefore, g is a fuzzy strong prime ideal. The proof is completed.

Theorem 32. Suppose that L is a prelinear coresiduated lattice; f is a fuzzy prime ideal and [alpha] [member of] (f(0), 1]; then f [conjunction] [alpha] is a fuzzy prime ideal of L where

(f [conjunction] [alpha])(x) = f(x) [conjunction] [alpha]. (20)

Proof. Since f is a fuzzy prime ideal, then f is a fuzzy ideal. [for all]x, y, z [member of] L, if z [less than or equal to] x [direct sum] y, then f(z) [less than or equal to] f(x) [disjunction] f(y). So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Thus f [conjunction] [alpha] is a fuzzy ideal. By [alpha] [member of] (f(0),1],it is obtained that f [conjunction] [alpha] is not a constant and (f [conjunction] [alpha])(0) = f(0) [conjunction] [alpha] = f(0), (f [conjunction] [alpha])(x) = f(x) [conjunction] [alpha] [less than or equal to] f(x). Since f is a fuzzy prime ideal, then according to Theorem 31 f [conjunction] [alpha] is a fuzzy prime ideal of L.

Theorem 33. Suppose that L is a prelinear coresiduated lattice and f is a fuzzy ideal and not a constant satisfying f(0) > 0; then there exists a fuzzy prime ideal g such that g [less than or equal to] f.

Proof. Since f is a fuzzy ideal and not a constant, it follows that [there exists]a [member of] L, s.t. f(a) [not equal to] f(0); that is, a [bar.[member of]] [[OMEGA].sub.f]. So [[OMEGA].sub.j] is a real ideal of L. It follows from Theorem 26 that there exists a prime ideal I such that [[OMEGA].sub.J] [subset or equal to] I and a [bar.[member of]] I. Thus [X.sub.I] is a fuzzy prime ideal. Put [alpha] = inf [f(x) | x [member of] L\ I}; then [alpha] [greater than or equal to] f(0) > 0. So it follows from Theorem 32 that [X.sub.I] [conjunction] [alpha] is a fuzzy prime ideal.

Let g = [X.sub.I] [conjunction] [alpha]; then x [member of] I, g(x) = ([X.sub.I](x)) [conjunction] [alpha] = 0 [conjunction] [alpha] = 0 < f(0) [less than or equal to] f(x); x [bar.[member of]] I, g(x) = ([X.sub.I](x)) [conjunction] [alpha] = 1 [conjunction] [alpha] = [alpha] = inf [f(x) | x [member of] L \ 1} [less than or equal to] f(x). It follows from the above that g [less than or equal to] f. The proof is completed.

The proof of the above theorem indicates how to obtain a fuzzy prime ideal derived from a fuzzy ideal in a prelinear coresiduated lattice.

6. Conclusion

Coresiduated lattice, as a dual algebra structure of residuated lattice, provides a new frame for logic algebras. It is a useful tool to characterize the intuitionistic fuzzy operators and plays a vital role in the theory basses of Triple I method of intuitionistic fuzzy reasoning. It is well known that ideal is an important part of algebra structure for various fuzzy logic semantics. From the point of view of fuzzy set, it can be characterized by using the notion of fuzzy ideal. In the paper, we mainly introduce the concepts of fuzzy ideal, fuzzy prime ideal, and fuzzy strong prime ideal to coresiduated lattices and derive some of their characterizations. We explore the relationship between fuzzy ideals and ideals in the coresiduated lattice as well. Moreover, we investigate the properties of fuzzy ideal and fuzzy (strong) prime ideal in prelinear coresiduated lattices.

The research on the relation between the proposed fuzzy ideals of coresiduated lattices and fuzzy ideals/filters of residuated lattices is worth doing. The detailed discussion about it will be given in our future work.

http://dx.doi.org/10.1155/2016/6423735

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors acknowledge their supports from the Natural Science Foundation of China (nos. 61473336, 11401361, and 61572016) and the Fundamental Research Funds for the Central Universities (no. GK201403001).

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Yan Liu (1) and Mucong Zheng (2)

(1) College of Science, Xi'an University of Science and Technology, Xi'an 710054, China

(2) School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, China

Correspondence should be addressed to Mucong Zheng; zhengmucong@aliyun.com

Received 27 January 2016; Revised 30 June 2016; Accepted 3 July 2016

Academic Editor: Remi Leandre
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