Characterizations of Fuzzy Ideals in Coresiduated Lattices

The notions of fuzzy ideals are introduced in coresiduated lattices. The characterizations of fuzzy ideals, fuzzy prime ideals, and fuzzy strong prime ideals in coresiduated lattices are investigated and the relations between ideals and fuzzy ideals are established. Moreover, the equivalence of fuzzy prime ideals and fuzzy strong prime ideals is proved in prelinear coresiduated lattices. Furthermore, the conditions under which a fuzzy prime ideal is derived from a fuzzy ideal are presented in prelinear coresiduated lattices.


Introduction
Residuated lattices provide an algebra frame for the algebraic semantics of formal fuzzy logics such as MV-algebras, BLalgebras, and  0 -algebras (or MTL-algebras) [1][2][3][4][5][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][8][9][10][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 [3].Actually, the original definition of MV-algebra was characterized by the operator ⊕ which could be looked as the generation of conorm in lattices.Obviously, there existed an operator ⊖ such that (⊕, ⊖) constituted a pair just like adjoint pair (⊗, → ).Therefore, the definition of coadjoint pair was presented in [12], and the definition of coresiduated lattice, as the dual algebra structure of residuated lattice, was introduced in [12].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 [13].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 [14].The properties of ideals in coresiduated lattices were investigated and the embedding theorem of coresiduated lattices was obtained in [15].
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 [18].Zhu and Xu extended the fuzzy filters in BL-algebras and MTLalgebras to general residuated lattices [19].
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.

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

Advances in Mathematical Physics
Definition 1 (see [12]).Let  be a poset.(⊕, ⊖) is called a coadjoint pair on  if the following conditions are satisfied.
⊕ is called a -conorm operator on , and ⊖ is called a fuzzy difference operator on .One denotes CRL as the set of all coresiduated lattices.
Theorem 6 (see [15]).Suppose  ∈ . is a real ideal of  and  is a (strong) prime ideal of ; if  ⊆ , then  is a (strong) prime ideal of .
Proposition 7 (see [15]).Consider  ∈ ; if  is a strong prime ideal of , then  is a prime ideal of .

Fuzzy Ideals of Coresiduated Lattices
Definition 8. Suppose that  is a fuzzy set on ,  ∈ CRL. is a fuzzy ideal of  if the following conditions are satisfied: Throughout the rest of this paper, unless otherwise stated,  always represents a coresiduated lattice and  always represents a fuzzy set on .Proposition 9. Consider  ∈  and  is a fuzzy ideal of ; then f is isotone; that is,
From Theorem 10, it is easy to get the following corollary.
Corollary 11.Consider  ∈  and  is a fuzzy set on ; then the following conditions are equivalent: (1)  is a fuzzy ideal of ; that is, (P1) and (P2) hold.
Corollary 12. Consider  ∈  and  is a fuzzy ideal on  if and only if Proof. Sufficiency.
Thus  is a fuzzy ideal on .
Let  be a fuzzy set on the coresiduated lattice ; it is denoted that is an ideal of .
Corollary 16.Suppose that  is a fuzzy ideal of ; then is an ideal of .

Fuzzy Prime Ideals and Fuzzy Strong Prime Ideals of Coresiduated Lattices
Definition 18. Suppose that  is a fuzzy ideal of  and not a constant. is a fuzzy prime ideal of  if the following conditions are satisfied: (7) ( ∧ ) = () ∧ (), ∀,  ∈ .
In this section, it is always assumed that  is a fuzzy set on  and not a constant.

Theorem 19. 𝑓 is a fuzzy prime ideal of 𝐿 if and only if one has the following: if 0 ̸
=   ⫋ , then   is a prime ideal of .
Necessity.Since  is a fuzzy strong prime ideal of , it is obvious that  is a fuzzy ideal of .According to Theorem 14, ∀ ∈ [0, 1], if   is an ideal of , then 0 ∈   ; that is, (0) ≤ .Moreover,  is a fuzzy strong prime ideal of .
Theorem 22.If  is a fuzzy strong prime ideal of  then  is a fuzzy prime ideal of .
Suppose that  is a real ideal of ; put According to Theorem 14,   is a fuzzy ideal and not a constant.By Theorems 19 and 21, we can get the following two theorems, respectively.
Theorem 23.  is a prime ideal of  if and only if   is a fuzzy prime ideal of .
Theorem 24. is a strong prime ideal of  if and only if   is a fuzzy strong prime ideal of .

Fuzzy Ideals and Fuzzy Prime Ideals of Prelinear Coresiduated Lattices
In a residuated lattice , the condition is called prelinearity axiom (see [1]).Taking account of coresiduated lattice as the dual algebra structure of residuated lattice, the definition of prelinear coresiduated lattice is as follows.
Theorem 26 (see [15]).Suppose that  is a prelinear coresiduated lattice;  is an ideal of  and  ∈  \ ; then there exists a prime ideal  of  such that  ⊆  and  .
Theorem 27.  is a prelinear coresiduated lattice. is a fuzzy strong prime ideal of  if and only if  is a fuzzy prime ideal of .
Proof.Necessity is obvious by Theorem 22.
Corollary 28.Suppose that  is a prelinear coresiduated lattice and  is a fuzzy ideal of ; then the following conditions are equivalent: (1)  is a fuzzy prime ideal of .
(3)  is a fuzzy strong prime ideal of .
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  is a prelinear coresiduated lattice;  is a fuzzy ideal and not a constant; then the following conditions are equivalent: (1)  is a chain.
(2) Any fuzzy ideal which is not a constant is a fuzzy prime ideal.
(3) Any fuzzy ideal such that (0) = 0 and not a constant is a fuzzy prime ideal.
Theorem 33.Suppose that  is a prelinear coresiduated lattice and  is a fuzzy ideal and not a constant satisfying (0) > 0; then there exists a fuzzy prime ideal  such that  ≤ .
The proof of the above theorem indicates how to obtain a fuzzy prime ideal derived from a fuzzy ideal in a prelinear coresiduated lattice.

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