On ideals of residuated lattices

1 Introduction

It is generally known that many-valued logic can be used to process various uncertain information as a powerful tool. Since the Basic Logic was first proposed by P. Hájek [7], BL-algebras have been widely studied as the algebraic structure of truth degrees related to this logic systems. Subsequently, based on the well known statement that a t-norm has residuum iff the t-norm remains left-continuous, Esteva and Godo [5] introduced a more general many-valued logic than Basic Logic, which is called Monoidal t-norm Based Logic (MTL for short). And the algebraic structure of MTL is called MTL-algebras, which plays an indispensable role in verifying the completeness of the Monoidal t-norm Based Logic. Both BL-algebras and MTL-algebras are the special classes of residuated lattices [21], and BL-algebras are the special class of MTL-algebras. MV-algebras, Π-algebras and R₀-algebras are also the most known classes of residuated lattices [5, 19]. Recently, many researchers investigated residuated lattices from different aspects, including topological algebras [22, 23], state [8], derivation [9] and very true operators [20].

The theory of filters plays an indispensable role in studying logic algebras because filters can be seen as the set of provable formulas. In logical algebra, the notion of ideals appeared in MV-algebras and lattice implication algebras. In fact, for any filter F of an MV-algebra M, F^∗ = {x^∗|x ∈ F} is an ideal of M, which means that the duality of a filter is an ideal in MV-algebras. On the contrary, the duality of an ideal is a filter in MV-algebras. This implies we only need to choose one of the filters and the ideals to study in MV-algebras. However, Lele and Nganou [11] extended the concept of ideal given in MV-algebras to BL-algebras and they constructed a series of examples to show the duality of a filter may not be an ideal in BL-algebras. Furthermore, they stated that the behavior of ideals is very different from filters in BL-algebras and the notion of ideals has a proper meaning in BL-algebras from a purely algebraic point of view. In 2017, Liu etl., [12] builded the theory of ideals on general residuated lattices. But there exist some mistakes in [12]. Especially Theorem 3.9 in [12] gave the wrong ideal generation formula on residuated lattices. By these motivations, we will try to rectify Theorem 3.9 in [12] and give the correct ideal generation formula on residuated lattices.

Based on the concept of ideals given in Lele and Nganou [11], Zou and Xin [24] introduced the concepts of annihilators and α-ideals in BL-algebras and mainly focused on studying the structure I _α  (M) of all α-ideals in a linear BL-algebra M. But in this paper, we will prove the set I _α  (M) contains only two trivial α-ideals {0} and M, when M is a linear BL-algebra. It implies that the set I _α  (M) is a trivial structure. Thus some results related to I _α  (M) mentioned in [24] are trivial. This inspires us to investigate some properties of I _α  (M) on non-linear logical algebras. Therefore, we study the structure of I _α  (M) on general MTL-algebras and get some non-trivial results. In addition, we extend the notion of annihilators to MTL-algebras, it is shown that an annihilator is an ideal on an MTL-algebra, but this may not hold in residuated lattice.

2 Preliminaries

We first give some existing concepts and notations on residuated lattices.

Definition 2.1. ([1, 7]) (i) We say an algebraic structure (A, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice provided that

(a) (A, ∧ , ∨ , 0, 1) forms a bounded lattice,

(b) (A, ⊙ , 1) is a commutative monoid,

(c) δ ⊙ ξ ≤ θ iff δ ≤ ξ → θ, for all δ, ξ, θ ∈ A.

(ii) A residuated lattice A is called an MTL-algebra provided that (δ → ξ) ∨ (δ → ξ) =1 for all δ, ξ ∈ A.

(iii) An MTL-algebra is called a BL-algebra provided that δ ∧ ξ = δ ⊙ (δ → ξ) for all δ, ξ ∈ A.

Let us denote δ → 0 by δ^∗, then we have the following properties.

Proposition 2.2 ([1, 16]) The following properties (R1)-(R15) hold on residuated lattices, (R1)-(M5) hold on MTL-algebras and (R1)-(B2)hold in BL-algebras.

δ ≤ ξ ⇔ δ → ξ = 1.

The authors [12] defined the operation ⊕ on a residuated lattice A as: δ ⊕ ξ = δ^* → ξ.

Definition 2.3. ([12]) Assume A is a residuated lattice and ∅ ≠ I ⊆ A. We say I is an ideal provided that for any δ, ξ ∈ A,

(I1) δ ≤ ξ, ξ ∈ I ⇒ δ ∈ I,

(I2) δ, ξ ∈ I ⇒ δ ⊕ ξ ∈ I.

We say an ideal I is a proper ideal on a residuated lattice A provided that I ≠ A.

Definition 2.4. ([2]) Assume L is a lattice with the minimum element 0 and δ ∈ L. If there is an element ξ ∈ L satisfying δ ∧ ξ = 0 and for each θ ∈ L, θ ∧ δ = 0 implies θ ≤ ξ, then we say ξ is the pseudo-complement of δ. A lattice is called a pseudo-complemented lattice provided that each element has a pseudo-complement.

3 Notes on ideals and fuzzy ideals on residuated lattices

In this part, we first pointed out some mistakes in [12] and further rectify the ideal generation formula given in [12]. Also we give the correct ideal generation formula on residuated lattices. Finally, we give some properties of ideals and discuss the structure of all ideals on MTL-algebras.

Notation. In [12], some properties with respect to the operation ⊕ were given on a residuated lattice A. But there exist some mistakes in [12]. For example, the Proposition 2.3 in [12] gave that (P21): δ ⊕ δ^∗ = 0, (P23): (δ ∧ ξ) ⊕ θ = (δ ⊕ θ) ∧ (ξ ⊕ θ) and (P25): (δ ⊕ ξ) ⊕ θ = δ ⊕ (ξ ⊕ θ), we observe that these properties are not correct in general. (P21) is incorrect by the fact that δ ⊕ δ^∗ = δ^∗ → δ^∗ = 1. And we can take the following example to indicate that (P23) and (P25) may not be true on residuated lattices.

Example 3.1. Let A = {0, δ, ξ, θ, γ, 1} and ⊙, → be given by:

OPEN IN VIEWER

Fig. 1

The diagram of Example 3.1.

Then (A, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice. We can find (ξ ∧ θ) ⊕ γ = (ξ ∧ θ) ^∗ → γ = 1 → γ = γ, but ( ξ ⊕ γ ) ∧ ( θ ⊕ γ ) = ( ξ ∗ → γ ) ∧ ( θ ∗ → γ ) = ( θ → γ ) ∧ ( ξ → γ ) = δ ∧ θ = θ . Thus (ξ ∧ θ) ⊕ γ ≠ (ξ ⊕ γ) ∧ (θ ⊕ γ). This means that (P23) may not be true on residuated lattices. We also find ξ ⊕ (ξ ⊕ γ) = ξ^∗ → (ξ^∗ → γ) = θ → (θ → γ) = θ → δ = 1, but ( ξ ⊕ ξ ) ⊕ γ = ( ξ ⊕ ξ ) ∗ → γ = ( ξ ∗ → ξ ) ∗ → γ = ( θ → ξ ) ∗ → γ = ξ ∗ → γ = θ → γ = δ .

Although the statement (P23): (δ ∧ ξ) ⊕ θ = (δ ⊕ θ) ∧ (ξ ⊕ θ) is false on residual lattices, the following proposition shows that it holds on MTL-algebras. For instance, we give some other properties of ideals on residuated lattices in the following.

Proposition 3.2. Let A be a residuated lattice. Then for any δ, ξ, θ ∈ A,

δ ≤ ξ ⇒ δ ⊕ θ ≤ ξ ⊕ θ,  θ ⊕ δ ≤ θ ⊕ ξ,

Proof. The proofs of (i), (ii) and (iii) are omitted.

(iv). Since ( δ ⊕ ξ ) → ( δ ∗ ⊙ ξ ∗ ) ∗ = ( δ ∗ ⊙ ξ ∗ ) → ( δ ⊕ ξ ) ∗ = ( δ ∗ ⊙ ξ ∗ ) → ( δ ∗ → ξ ) ∗ = δ ∗ → ( ξ ∗ → ( δ ∗ → ξ ) ∗ ) ≥ δ ∗ → ( ( δ ∗ → ξ ) → ξ ) = 1 , which implies δ ⊕ ξ ≤ (δ^∗ ⊙ ξ^∗) ^∗. Thus (δ ⊕ ξ) ^∗ ≥ (δ^∗ ⊙ ξ^∗) ^∗∗ ≥ δ^∗ ⊙ ξ^∗. (v). By (iv) and Proposition 2.2 (R8), we have ( δ ⊕ ξ ) ⊕ θ = ( δ ⊕ ξ ) ∗ → θ ≤ ( δ ∗ ⊙ ξ ∗ ) → θ = δ ∗ → ( ξ ∗ → θ ) = δ ⊕ ( ξ ⊕ θ ) . (vi). By Proposition 2.2 (R5), we have δ ⊕ ( ξ ∧ θ ) = δ ∗ → ( ξ ∧ θ ) = ( δ ∗ → ξ ) ∧ ( δ ∗ → θ ) = ( δ ⊕ ξ ) ∧ ( δ ⊕ θ ) .

(vii). By Proposition 2.2 (M1)(R4) and Proposition 3.2 (vi), we have ( δ ∧ ξ ) ⊕ ( δ ∧ θ ) = ( δ ∧ ξ ) ∗ → ( δ ∧ θ ) = ( δ ∗ ∨ ξ ∗ ) → ( δ ∧ θ ) = ( δ ∗ → ( δ ∧ θ ) ) ∧ ( ξ ∗ → ( δ ∧ θ ) ) = ( δ ⊕ ( δ ∧ θ ) ) ∧ ( ξ ⊕ ( δ ∧ θ ) ) = ( δ ⊕ δ ) ∧ ( δ ⊕ θ ) ∧ ( ξ ⊕ δ ) ∧ ( ξ ⊕ θ ) ≥ δ ∧ δ ∧ δ ∧ ( ξ ⊕ θ ) = δ ∧ ( ξ ⊕ θ ) .

(viii). By Proposition 2.2 (M1)(R4), we have ( δ ∧ ξ ) ⊕ w = ( δ ∧ ξ ) ∗ → w = ( δ ∗ ∨ ξ ∗ ) → w = ( δ ∗ → w ) ∧ ( ξ ∗ → w ) = ( δ ⊕ w ) ∧ ( ξ ⊕ w ) . □

Notation. The Remark 2.4 in [11] showed that ⊕ is not commutative in BL-algebras. By the fact that BL-algebras are the particular class of MTL-algebras, we get that ⊕ is also not commutative on MTL-algebras. Furthermore, the following example shows that ⊕ is non-associative on MTL-algebras.

Example 3.3. Assume M = {0, δ, ξ, θ, 1} is a chain and the operation ⊙, → are given by

Then (M, ∧ , ∨ , ⊙ , → , 0, 1) is an MTL-algebra. We can find (δ ⊕ ξ) ⊕0 = (δ^∗ → ξ) ^∗ → 0 = δ → 0 = θ and δ ⊕ (ξ ⊕ 0) = δ^∗ → (ξ^∗ → 0) = θ → θ = 1. This implies (δ ⊕ ξ) ⊕0 ≠ δ ⊕ (ξ ⊕ 0). Therefore, ⊕ is non-associative on MTL-algebras.

Notation. Let A be a residuated lattice and ∅ ≠ X ⊆ A, then the symbol 〈X〉 means 〈X〉 is the least ideal containing X. The Theorem 3.9 in [12] showed 〈X〉 = {a ∈ A|a ≤ (·· · ((x₁ ⊕ x₂) ⊕ x₃) ⊕ ·· ·) ⊕ x _n , x _i  ∈ X}. But we can take the following example to indicate that the ideal generation formula of Theorem 3.9 in [12] is false.

Example 3.4. Let A = 〈[0, 1] , ∧ , ∨ , ⊙ , → , 0, 1〉 such that, for any δ, ξ ∈ [0, 1], δ ⊙ ξ = { 0 , if δ , ξ ∈ [ 0 , 1 2 ] m i n { δ , ξ } , elsewhere δ → ξ = { 1 , if δ ≤ ξ 1 2 , if ξ < δ ≤ 1 2 ξ , if ξ < δ , 1 2 < δ .

Then 〈[0, 1] , ∧ , ∨ , ⊙ , → , 0, 1〉 is a residuated lattice [17]. Let X = { 1 3 } . According to the ideal generation formula in [12], there exists n ∈ ℕ such that 〈 X 〉 = { a ∈ A | a ≤ ( · · · ( ( 1 3 ⊕ 1 3 ) ⊕ 1 3 ) ⊕ · · · ) ⊕ 1 3 } . Since 1 3 ⊕ 1 3 = ( 1 3 ) ∗ → 1 3 = 1 2 → 1 3 = 1 2 , we have ( 1 3 ⊕ 1 3 ) ⊕ 1 3 = ( 1 2 ) ∗ → 1 3 = 1 2 → 1 3 = 1 2 . By induction, we get that (·· · ( ( 1 3 ⊕ 1 3 ) ⊕ 1 3 ) ⊕ · · · ) ⊕ 1 3 = 1 2 . Thus 〈 X 〉 = [ 0 , 1 2 ] . But 1 3 ⊕ ( 1 3 ⊕ 1 3 ) = ( 1 3 ) ∗ → 1 2 = 1 2 → 1 2 = 1 . Clearly 1 ∉ 〈X〉, which is a contradiction to the definition of ideals. Therefore, the ideal generation formula of Theorem 3.9 in [12] is not true.

In the following theorem, we give the correct method to generate an ideal on residuated lattices.

Theorem 3.5. Assume A is a residuated lattice and ∅ ≠ X ⊆ A. Then 〈X〉 = {a ∈ A |  ∃ δ₁, δ₂, . . . , δ _n  ∈ X s . t .  a ≤ δ _n  ⊕ (. . . ⊕ (δ₃ ⊕ (δ₂ ⊕ δ₁)) . . .)} .

Proof. Let U = {a ∈ A |  ∃ δ₁, δ₂, . . . , δ _n  ∈ X s . t .  a ≤ δ _n  ⊕ (. . . ⊕ (δ₃ ⊕ (δ₂ ⊕ δ₁)) . . .)} . Clearly, 0 ∈ U and X ⊆ U. If b ≤ a and a ∈ U, then b ∈ U clearly.

Now, let a, b ∈ U. Then there exist n , m ∈ ℕ and δ₁, δ₂, . . . , δ _n  ∈ X, ξ₁, ξ₂, . . . , ξ _m  ∈ X such that a ≤ δ _n  ⊕ (·· · ⊕ (δ₃ ⊕ (δ₂ ⊕ δ₁)) ·· ·) and b ≤ ξ _m  ⊕ (·· · ⊕ (ξ₃ ⊕ (ξ₂ ⊕ ξ₁)) ·· ·) . Thus by Proposition 3.2 (i) (v), we get a ⊕ b ≤ [ δ n ⊕ ( · · · ⊕ ( δ 3 ⊕ ( δ 2 ⊕ δ 1 ) ) · · · ) ] ⊕ [ ξ m ⊕ ( · · · ⊕ ( ξ 3 ⊕ ( ξ 2 ⊕ ξ 1 ) ) · · · ) ] ≤ δ n ⊕ { [ δ n - 1 ⊕ ( · · · ⊕ ( δ 2 ⊕ δ 1 ) · · · ) ] ⊕ [ ξ m ⊕ ( · · · ⊕ ( ξ 3 ⊕ ( ξ 2 ⊕ ξ 1 ) ) · · · ) ] } ≤ δ n ⊕ ( δ n - 1 ⊕ { [ δ n - 2 ⊕ ( · · · ⊕ ( δ 2 ⊕ δ 1 ) · · · ) ] ⊕ [ ξ m ⊕ ( · · · ⊕ ( ξ 2 ⊕ ξ 1 ) · · · ) ] } ) ⋮ ≤ δ n ⊕ ( δ n - 1 ⊕ ( · · · ( δ 2 ⊕ ( δ 1 ⊕ ( ξ m ⊕ ( · · · ⊕ ( ξ 2 ⊕ ξ 1 ) · · · ) ) ) ) · · · ) ) . Hence a ⊕ b ∈ U. Therefore, U is an ideal containing X.

For any ideal I with X ⊆ I, take a ∈ U, then there exist n ∈ ℕ and δ₁, δ₂, . . . , δ _n  ∈ X satisfying a ≤ δ _n  ⊕ (·· · ⊕ (δ₃ ⊕ (δ₂ ⊕ δ₁)) ·· ·), this means a ∈ I. Thus U ⊆ I. Therefore, U is the smallest ideal containing X, i.e. 〈X〉 = U. □

Assume A is a residuated lattice and δ ∈ A, we say 〈 {δ} 〉 is the principal ideal of A, denoted by 〈δ〉. Thus we have 〈δ〉 = {ξ ∈ A|ξ ≤ nδ}, where n δ = δ ⊕ ( · · · ⊕ ( δ ⊕ ( δ ⊕ δ ) ) · · · ) ︸ n times .

Remark 3.6. In [11], the authors showed that for any nonempty subset X of a BL-algebra M, 〈X〉 = {a ∈ M|a ≤ (·· · ((δ₁ ⊕ δ₂) ⊕ δ₃) ⊕ ·· ·) ⊕ δ _n , δ _i  ∈ X, i = 1, 2 ·· · , n} . From [11] we get that ⊕ is associative in BL-algebras, so the ideal generation formula in the above theorem and the ideal generation formula given by [11] are consistent in BL-algebras.

Let us use I (M) to represent the set of all ideals on an MTL-algebra M.

Proposition 3.7. Assume M is an MTL-algebra. Then for arbitrary δ, ξ ∈ M,

if δ ≤ ξ, then 〈δ〉 ⊆ 〈ξ〉,

Proof. (i) Take z ∈ 〈δ〉. Then there exists n ∈ ℕ such that z ≤ nδ ≤ nξ, which implies z ∈ 〈ξ〉.

(ii) From δ, ξ ≤ δ ⊕ ξ and (i), we have 〈δ〉, 〈ξ〉 ⊆ 〈δ ⊕ ξ〉. This implies that 〈δ ⊕ ξ〉 is an upper bound of 〈δ〉 and 〈ξ〉 in I (M). For any I ∈ I (M) satisying 〈δ〉 ⊆ I and 〈ξ〉 ⊆ I, we get δ, ξ ∈ I, which implies δ ⊕ ξ ∈ I. Since I is an ideal, we have 〈δ ⊕ ξ〉 ⊆ I. Therefore, 〈δ〉 ∨  _I 〈ξ〉 = 〈δ ⊕ ξ〉.

(iii) Since δ ∧ ξ ≤ δ, ξ, we have 〈δ ∧ ξ〉 ⊆ 〈δ〉 ∩ 〈ξ〉. Conversely, let t ∈ 〈δ〉 ∩ 〈ξ〉. Then there exist n , m ∈ ℕ satisfying t ≤ nδ and t ≤ mξ. Let q = max {n, m}. Then t ≤ qδ, qξ. Now, we prove n (δ ∧ ξ) = ⋀ {a _n  ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·) |a _i  = δ or a _i  = ξ}. When n = 2, by Proposition 3.2 (vi) and (viii), we have 2 ( δ ∧ ξ ) = ( δ ∧ ξ ) ⊕ ( δ ∧ ξ ) = ( ( δ ∧ ξ ) ⊕ δ ) ∧ ( ( δ ∧ ξ ) ⊕ ξ ) = ( δ ⊕ δ ) ∧ ( ξ ⊕ δ ) ∧ ( δ ⊕ ξ ) ∧ ( ξ ⊕ ξ ) = ⋀ { a 1 ⊕ a 2 | a i = δ or ξ } . Assume the equation n (δ ∧ ξ) = ⋀ {a _n  ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·) |a _i  = δ or ξ} holds when k = n. Now, let k = n + 1. Then we get ( n + 1 ) ( δ ∧ ξ ) = ( δ ∧ ξ ) ⊕ ( n ( δ ∧ ξ ) ) = ( δ ∧ ξ ) ⊕ ⋀ { a n ⊕ ( ⋯ ⊕ ( a 2 ⊕ a 1 ) ⋯ ) | a i = δ or ξ } = ( δ ⊕ ⋀ { a n ⊕ ( ⋯ ⊕ ( a 2 ⊕ a 1 ) ⋯ ) | a i = δ or ξ } ) ∧ ( ξ ⊕ ⋀ { a n ⊕ ( ⋯ ⊕ ( a 2 ⊕ a 1 ) ⋯ ) | a i = δ or ξ } ) = ⋀ { δ ⊕ ( a n ⊕ ( ⋯ ⊕ ( a 2 ⊕ a 1 ) ⋯ ) ) | a i = δ or ξ } ∧ ⋀ { ξ ⊕ ( a n ⊕ ( ⋯ ⊕ ( a 2 ⊕ a 1 ) ⋯ ) ) | a i = δ or ξ } = ⋀ { a n + 1 ⊕ ( a n ⊕ ( ⋯ ⊕ ( a 2 ⊕ a 1 ) ⋯ ) ) | a i = δ or ξ } . Therefore, 2q (δ ∧ ξ) = ⋀ {a_2q ⊕ (·· · ⊕ (a₂ ⊕ a₁) ·· ·) |a _i  = δ or ξ}. Now, we show a_2q ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·) ≥ qδ or qξ. The reason is as follows. Since for any 1 ≤ i ≤ 2q, a _i  = δ or ξ, then there exist two cases:

(1). If the number of δ among a₁, a₂ ·· · a_2q is less than q, then the number of ξ among a₁, a₂ ·· · a_2q is greater than q. By Proposition 3.2(i) and (ii), we have a_2q ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·) ≥ qξ.

(2). If the number of δ among a₁, a₂ ·· · a_2q is greater or equal to q, then the number of ξ among a₁, a₂ ·· · a_2q is less than q. By Proposition 3.2 (ii), we have a_2q ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·) ≥ qδ.

Therefore, a_2q ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·) ≥ qδ or qξ. Hence, ⋀ {a_2q ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·) |a _i  = δ or ξ} ≥ qδ ∧ qξ ≥ t ∧ t = t . It results that t ≤ 2q (δ ∧ ξ), which implies t ∈ 〈δ ∧ ξ〉. Therefore, 〈δ〉 ∩ 〈ξ〉=〈δ ∧ ξ〉. □

Notation The Proposition 3.7 (iii) may not be true on residuated lattices. For example, in Example 4.3, 〈γ〉 ∩ 〈θ〉 = {0, γ, η} ∩ M = {0, γ, η}. But 〈γ ∧ θ〉 = 〈0〉 = {0}, so 〈γ〉 ∩ 〈θ〉 ≠ 〈γ ∧ θ〉.

In the following theorem, we will discuss the lattice structure of I (M).

Theorem 3.8. Assume M is an MTL-algebra, then (I (M) , ∩ , ∨  _I , {0} , M) is a distributive lattice, where I ∨  _I J = 〈I ∪ J〉 for all I, J ∈ I (M).

Proof. It is trivial that (I (M) , ∩ , ∨  _I , {0} , M) is a bounded lattice. Next, we prove that the distributive law holds, i.e. for all I, J, Q ∈ I (M), I ∩ ( J ∨ I Q ) = ( I ∩ J ) ∨ I ( I ∩ Q ) . In fact, (I ∩ J) ∨  _I  (I ∩ Q) ⊆ I ∩ (J ∨  _I Q) obviously holds. On the other hand, let δ ∈ I ∩ (J ∨  _I Q). then we have δ ∈ I and δ ∈ J ∨  _I Q. It gives that δ ≤ a _n  ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·), where a _i  ∈ J or Q. Since I, J, Q ∈ I (M), it implies δ ∧ a _i  ∈ I ∩ J or I ∩ Q. Now, we prove δ ≤ (δ ∧ a _n ) ⊕ (·· · ⊕ ((δ ∧ a₃) ⊕ ((δ ∧ a₂) ⊕ (δ ∧ a₁))) ·· ·). Since δ ≤ a _n  ⊕ (·· · ⊕ (a₃ ⊕ (a₂ ⊕ a₁)) ·· ·), by Proposition 3.2 (vii) we have δ = δ ∧ ( a n ⊕ ( · · · ⊕ ( a 3 ⊕ ( a 2 ⊕ a 1 ) ) · · · ) ) ≤ ( δ ∧ a n ) ⊕ ( · · · ⊕ ( ( δ ∧ a 2 ) ⊕ ( δ ∧ a 1 ) ) · · · ) , which implies δ ∈ (I ∩ J) ∨  _I  (I ∩ Q). Hence, I ∩ (J ∨  _I Q) = (I ∩ J) ∨  _I  (I ∩ Q). Therefore, (I (M) , ∩ , ∨  _I , {0} , M) is a distributive lattice. □

In this part, we mainly introduce the concept of annihilators on MTL-algebras and we show an annihilator is an ideal on MTL-algebras, but an example can be used to indicate that an annihilator may not be an ideal on residuated lattices.

In the following research, M represents an MTL-algebra unless otherwise specified.

Definition 4.1. Assume ∅ ≠ G ⊆ M. We say G ⊤ = { δ ∈ M | ∀ ξ ∈ G , δ ∧ ξ = 0 } , is an annihilator of G and for each δ ∈ M, we write {δ} ^⊤ as δ^⊤. It can be easily seen that M^⊤ = {0} and 0^⊤ = M.

Theorem 4.2. Assume ∅ ≠ G ⊆ M, then G^⊤ is an ideal.

Proof. Let δ ∈ G^⊤ and θ ≤ δ, then θ ∧ ξ ≤ δ ∧ ξ = 0 for all ξ ∈ G, which implies θ ∈ G^⊤. Suppose δ, θ ∈ G^⊤, so for arbitrary ξ ∈ G, ξ ∧ δ = ξ ∧ θ = 0. From Proposition 3.2 (vii), we get that ξ ∧ (δ ⊕ θ) ≤ (ξ ∧ δ) ⊕ (ξ ∧ θ) =0 ⊕ 0 =0. Hence (δ ⊕ θ) ∈ G^⊤. Therefore, G^⊤ is an ideal. □

Example 4.3. Assume M = {0, δ, ξ, θ, γ, 1} and ⊙,→ are defined as follows.

OPEN IN VIEWER

Fig. 2

The diagram of Example 4.3.

Then (M, ∧ , ∨ , ⊙ , → , 0, 1) is an MTL-algebra. It is immediately clear that {0, δ, ξ} ^⊤ = {0, θ}. And {0, θ} is an ideal of M.

However, the following example can indicate that the Theorem 4.2 doesn’t hold on residuated lattices in general.

Example 4.4. Assume A = {0, δ, ξ, θ, γ, e, 1}, with 0 < θ < ξ < δ < 1, 0 < γ < η < δ < 1 and {ξ, θ} are incomparable with {γ, η}. Define the operations ⊙ and → on A as follows:

OPEN IN VIEWER

Fig. 3

The diagram of Example 4.4.

Then (A, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice but it is not an MTL-algebra, because (γ → θ) ∨ (θ → γ) = δ ∨ η = δ ≠ 1. Let X = {0, γ}, then X^⊤ = {0, γ} ^⊤ = {0, θ, ξ}, but {0, θ, ξ} is not an ideal of A, because θ ⊕ ξ = θ^∗ → ξ = δ ∉ {0, θ, ξ}.

Proposition 4.5. For any nonempty subsets J, Q of M, then

J ⊆ Q ⇒ Q^⊤ ⊆ J^⊤,

Proof. We only prove (iv). The rest of the proofs can refer to [24].

(iv) Since J ⊆ 〈J〉, by (i) we have 〈J〉^⊤ ⊆ J^⊤. Conversely, let t ∈ J^⊤. Then δ ∧ t = 0, for each δ ∈ J. Let z ∈ 〈J〉. Then there exist n ∈ ℕ and δ₁, δ₂, δ₃, . . . , δ _n  ∈ J such that z ≤ δ _n  ⊕ (·· · ⊕ (δ₃ ⊕ (δ₂ ⊕ δ₁)) ·· ·), and t ∧ δ _i  = 0  (i = 1, 2, ·· · n). Now we prove that t ∧ ( δ n ⊕ ( · · · ⊕ ( δ 3 ⊕ ( δ 2 ⊕ δ 1 ) ) · · · ) ) = 0 . By Proposition 3.2 (vii), it is easy to get t ∧ ( δ n ⊕ ( ⋯ ⊕ ( δ 2 ⊕ δ 1 ) ⋯ ) ) ≤ ( t ∧ δ n ) ⊕ ( ( t ∧ δ n - 1 ) ⊕ ( ⋯ ⊕ ( ( t ∧ δ 2 ) ⊕ ( t ∧ δ 1 ) ) ⋯ ) ) . So t ∧ z ≤ t ∧ (δ _n  ⊕ (·· · ⊕ (δ₃ ⊕ (δ₂ ⊕ δ₁)) ·· ·)) ≤0 ⊕ (0 ⊕ (·· · ⊕ (0 ⊕ (0 ⊕ 0)) ·· ·)) =0. It implies t ∧ z = 0, thus t ∈ 〈J〉^⊤. Therefore, G^⊤ ⊆ 〈J〉^⊤. □

Proposition 4.6. For any δ, ξ ∈ M, then

δ ≤ ξ ⇒ ξ^⊤ ⊆ δ^⊤,

Proof. (i) It obviously holds.

(ii) Since δ ≤ δ ⊕ ξ and ξ ≤ δ ⊕ ξ, by (i) we know (δ ⊕ ξ) ^⊤ ⊆ δ^⊤ and (δ ⊕ ξ) ^⊤ ⊆ ξ^⊤, which implies (δ ⊕ ξ) ^⊤ ⊆ δ^⊤ ∩ ξ^⊤. Conversely, take z ∈ δ^⊤ ∩ ξ^⊤, then we have z ∈ δ^⊤ and z ∈ ξ^⊤, which gives z ∧ δ = 0 and z ∧ ξ = 0. By Proposition 3.2 (vii), we get z ∧ (δ ⊕ ξ) =0, thus z ∈ (δ ⊕ ξ) ^⊤. Therefore, δ^⊤ ∩ ξ^⊤ ⊆ (δ ⊕ ξ) ^⊤. In a similar way, we can prove that δ^⊤ ∩ ξ^⊤ = (δ ∨ ξ) ^⊤.

(iii) It is evident that 0^⊤ = M. Conversely, suppose δ^⊤ = M. Since 1 ∈ M, we have δ = 1 ∧ δ = 0.

(iv) Clearly, (δ ∧ ξ) ^⊤⊤ ⊆ δ^⊤⊤ ∩ ξ^⊤⊤ holds. Conversely, let x ∈ δ^⊤⊤ ∩ ξ^⊤⊤ and y ∈ (δ ∧ ξ) ^⊤, which implies y ∧ δ ∧ ξ = 0, we thus get y ∧ δ ∈ ξ^⊤. Since x ∈ ξ^⊤⊤, we have x ∧ y ∧ δ = 0, which implies x ∧ y ∈ δ^⊤. Since x ∈ δ^⊤⊤, we have x ∧ y ∧ x = 0. Hence x ∧ y = 0 for all y ∈ (δ ∧ ξ) ^⊤. This gives that x ∈ (δ ∧ ξ) ^⊤⊤. Therefore, (δ ∧ ξ) ^⊤⊤ = δ^⊤⊤ ∩ ξ^⊤⊤.  □

In the following, we first extend the concept of α-ideals introduced by Zou etl [24] to MTL-algebras. We show that there are no non-trivial α-ideal on linear MTL-algebras. Further we point out that some results obtained in [24] are trivial. Finally, we study the relations between annihilators and α-ideal on MTL-algebras.

Definition 5.1. We say the ideal I of M is an α-ideal provided that for each δ ∈ I, δ^⊤⊤ ⊆ I. And we say {0} and M are trivial α-ideals.

Example 5.2. In Example 4.3, we can see that {0, δ, ξ} and {0, θ} are α-ideals of M.

Proposition 5.3. If ξ, β ∈ M and ξ ∧ β = 0 implies ξ = 0 or β = 0, then M does not contain any non-trivial α-ideal.

Proof. Suppose I is a non-trivial α-ideal of M, then there is an element θ ∈ I with 0 ≠ θ. For each ξ ∈ θ^⊤, we have ξ ∧ θ = 0, so ξ = 0 since 0 ≠ θ. It implies θ^⊤ = {0}. By the fact that I is a proper ideal, we get θ^⊤⊤ = M ⊈ I, which contradicts the assumption. Therefore, M doesn’t contain any non-trivial α-ideal. □

Let us represent the collection of all α-ideals of an MTL-algebra M by I _α  (M). As a consequence of Proposition 5.3, we have the following remark.

Remark. The authors [24] considered the lattice structure of I _α  (M) when M was a linear BL-algebra. But from Proposition 5, we get I _α  (M) contains only two elements {0} and M in a linear BL-algebra, and hence it is a trivial structure. Unlike the trivial results obtained in [24], we will study the lattice structures of I _α  (M) on non-linear MTL-algebras. The following example shows that I _α  (M) contains more than two elements on some non-linear MTL-algebras. This shows that our research has more significance.

Example 5.4. Assume M = {0, δ, ξ, θ, γ, η, ϑ, 1} such that 0 < δ < ξ < θ < 1, 0 < γ < η < ϑ < 1, δ < e and ξ < ϑ. And the operations ⊙ and → are given as follows:

Then (M, ∧ , ∨ , ⊙ , → , 0, 1) is an MTL-algebra and I _α  (M) = {{0} , {0, γ} , {0, δ, ξ, θ} , M}, it is evident I _α  (M) contains more than two elements.

Theorem 5.5. Assume I ∈ I (M), then the least α-ideal containing I is ⋃_θ∈Iθ^⊤⊤.

Proof. Let G = ⋃ _θ∈Iθ^⊤⊤. It is sufficient to show that G is an ideal. Let δ ∈ G with ξ ≤ δ. By Proposition 4.6 (i) and Proposition 4.5 (i), we can get ξ^⊤⊤ ⊆ δ^⊤⊤. By the fact that δ ∈ G = ⋃ _θ∈Iθ^⊤⊤, there exists θ₀ ∈ I satisfying δ ∈ θ 0 ⊤ ⊤ . It follows from Proposition 4.5 (i) that δ ⊤ ⊤ ⊆ θ 0 ⊤ ⊤ . Thus ξ ∈ ξ ⊤ ⊤ ⊆ θ 0 ⊤ ⊤ , which implies ξ ∈ G. Let δ, ξ ∈ I. Then there exist θ₁, θ₂ ∈ I such that δ ∈ θ 1 ⊤ ⊤ and ξ ∈ θ 2 ⊤ ⊤ . It implies θ 1 ⊤ ⊆ δ ⊤ , θ 2 ⊤ ⊆ ξ ⊤ . By Proposition 4.6 (ii), we get ( θ 1 ⊕ θ 2 ) ⊤ = θ 1 ⊤ ∩ θ 2 ⊤ ⊆ δ ⊤ ∩ ξ ⊤ = ( δ ⊕ ξ ) ⊤ , which gives δ ⊕ ξ ∈ (δ ⊕ ξ) ^⊤⊤ ⊆ (θ₁ ⊕ θ₂) ^⊤⊤ . Since I is an ideal, we get θ₁ ⊕ θ₂ ∈ I. This implies δ ⊕ ξ ∈ G. Therefore, G is an ideal.

For δ ∈ G, there is an θ ∈ I satisfying δ ∈ θ^⊤⊤ ⊆ G, we can deduce that δ^⊤⊤ ⊆ θ^⊤⊤. Hence G is an α-ideal. It is easily proved that I ⊆ G. Suppose J is an arbitrary α-ideal containing I. Then for each δ ∈ G, there exists θ ∈ I such that δ ∈ θ^⊤⊤. Since J is an α-ideal and I ⊆ J, we have δ ∈ θ^⊤⊤ ⊆ J. We can conclude G ⊆ J. Therefore, ⋃_θ∈Iθ^⊤⊤ is the least α-ideal containing I. □

Next, we investigate the relations between α-ideals and annihilators on M. First, we can easily prove that for ∅ ≠ X ⊆ M, X^⊤ is an α-ideal.

Lemma 5.6. Assume I ∈ I (M). Then I^⊤ = ⋂ _w∈Iw^⊤.

Proof. For any θ ∈ I, I^⊤ ⊆ θ^⊤ holds by Proposition 4.5(i). It results that I^⊤ ⊆ ⋂ _θ∈Iθ^⊤. On the other hand, for each ξ ∈ ⋂ _θ∈Iθ^⊤, then ξ ∧ θ = 0 for all θ ∈ I. This implies ξ ∈ I^⊤. Therefore, I^⊤ = ⋂ _θ∈Iθ^⊤. □

Theorem 5.7. Assume I ∈ I (M). If I contains maximum element, then the least α-ideal containing I is I^⊤⊤.

Proof. To prove the statement holds, we only need to show ⋃_θ∈Iθ^⊤⊤ = I^⊤⊤. By Proposition 4.5(i), we get ⋃_θ∈Iθ^⊤⊤ ⊆ I^⊤⊤. Conversely, let δ ∈ I^⊤⊤. We first denote the maximum element by θ₀, so we have θ ≤ θ₀ for all θ ∈ I. It implies θ 0 ⊤ ⊆ θ ⊤ for all θ ∈ I, so ⋂ θ ∈ I θ ⊤ = θ 0 ⊤ . Thus by Lemma 5.6, I ⊤ = θ 0 ⊤ , which gives δ ∈ I ⊤ ⊤ = θ 0 ⊤ ⊤ . Therefore, I^⊤⊤ ⊆ ⋃ _θ∈Iθ^⊤⊤. □

Corollary 5.8. Assume I ∈ I (M). If M is finite, then the least α-ideal containing I is I^⊤⊤.

Proof. From Theorem 5.7, we can obtain this result. □

Theorem 5.9. I _α  (M) , ∩ , ∨  _α , * , {0} , M) is a pseudo-complemented lattice, for any I, G ∈ I _α  (M), I ∨  _α G = ⋃ _{δ∈I∨ I G}δ^⊤⊤ = ⋃ _{δ∈〈I∪G〉}δ^⊤⊤, I^* = I^⊤.

Proof. Take I, G ∈ I _α  (M), we note that I ∨  _α G = ⋃ _{δ∈I∨ I G}δ^⊤⊤. Clearly I, G ⊆ ⋃ _{δ∈I∨ I G}δ^⊤⊤. So ⋃_{δ∈I∨ I G}δ^⊤⊤ is an upper bound of I and G in I _α  (M). Consider H ∈ I _α  (M) such that I ⊆ H and G ⊆ H, then I ∨  _I G ⊆ H. It implies δ ∈ H for all δ ∈ I ∨  _I G. Since H is an α-ideal, we have δ^⊤⊤ ∈ H. This implies ⋃_{δ∈I∨ I G}δ^⊤⊤ ⊆ H, we conclude that I ∨  _α G = ⋃ _{δ∈I∨ I G}δ^⊤⊤.

Finally, take I ∈ I _α  (M), we show that I^* = I^⊤. Clearly I ∩ I^⊤ = {0}. Consider Q ∈ I _α  (M) such that I ∩ Q = {0}, then by Proposition 4.5 (vii), we have Q ⊆ I^⊤. Therefore, (I _α  (M) , ∩ , ∨  _α , {0} , M) is a pseudo-complemented lattice. □

Theorem 5.10. (I _α  (M) , ∩ , ∨  _α , {0} , M) forms a distributive lattice.

Proof. Let I, G, K ∈ I _α  (M). Then by Theorem 3.8, we have ( I ∨ α G ) ∩ ( I ∨ α K ) = ⋃ δ ∈ I ∨ I G δ ⊤ ⊤ ∩ ⋃ ξ ∈ I ∨ I K ξ ⊤ ⊤ = ⋃ θ ∈ ( I ∨ I G ) ∩ ( I ∨ I K ) θ ⊤ ⊤ = ⋃ θ ∈ I ∨ I ( G ∩ K ) θ ⊤ ⊤ = I ∨ α ( K ∩ G ) . Now, we prove the second equation holds. It is clear that ⋃ θ ∈ ( I ∨ I G ) ∩ ( I ∨ I K ) θ ⊤ ⊤ ⊆ ⋃ δ ∈ I ∨ I G δ ⊤ ⊤ ∩ ⋃ ξ ∈ I ∨ I K ξ ⊤ ⊤ . Conversely, for x ∈ ⋃ _{δ∈I∨ I G}δ^⊤⊤ ∩ ⋃ _{ξ∈I∨ I K}ξ^⊤⊤, then there exist δ ∈ I ∨  _I G, ξ ∈ I ∨  _I K such that x ∈ δ^⊤⊤ and x ∈ ξ^⊤⊤, which implies x ∈ δ^⊤⊤ ∩ ξ^⊤⊤ = (δ ∧ ξ) ^⊤⊤ by Proposition 4.6(iv). Since δ ∧ ξ ∈ (I ∨  _I G) ∩ (I ∨  _I K), we have x ∈ ⋃ _{θ∈(I∨ I G)∩(I∨ I K)}θ^⊤⊤. Hence ⋃ θ ∈ ( I ∨ I G ) ∩ ( I ∨ I K ) θ ⊤ ⊤ ⊇ ⋃ δ ∈ I ∨ I G δ ⊤ ⊤ ∩ ⋃ ξ ∈ I ∨ I K ξ ⊤ ⊤ . Thus, ⋃ δ ∈ I ∨ I G δ ⊤ ⊤ ∩ ⋃ ξ ∈ I ∨ I K ξ ⊤ ⊤ = ⋃ θ ∈ ( I ∨ I G ) ∩ ( I ∨ I K ) θ ⊤ ⊤ . Therefore, (I _α  (M) , ∩ , ∨  _α , {0} , M) is distributive lattice. □

Proposition 5.11. If for each I ∈ I(M), I has the maximum element, then (I _α  (M) , ∩ , ∨  _α , {0} , M) is a Boolean algebra.

Proof. From Theorem 5.9 and Theorem 5.10, we can obtain that (I _α  (M) , ∩ , ∨  _α , {0} , M) is a pseudo-complemented lattice and distributive lattice. Now, let I ∈ I _α  (M). Next, we show that I^⊤ is the complement of I ∈ I _α  (M). Clearly, I^⊤ ∩ I = 0 and I^⊤ ∨  _α I = ⋃ _{δ∈I∨ I I^⊤}δ^⊤⊤. We note that 1 ∈ ⋃ _{δ∈I∨ I I^⊤}δ^⊤⊤. Let ξ be the maximum element of I ∨  _I I^⊤, then by Lemma 5.6 we have (I ∨  _I I^⊤) ^⊤ = ⋂ _{θ∈I∨ I I^⊤}θ^⊤ = ξ^⊤. Since (I ∨  _I I^⊤) ^⊤ ⊆ I^⊤ ∩ I^⊤⊤ = {0}, we get ξ^⊤ = {0}. It implies ξ^⊤⊤ = M, thus I^⊤ ∨  _α I = M. Therefore, (I _α  (M) , ∩ , ∨  _α , {0} , M) is a Boolean algebra. □

Corollary 5.12. Assume M is a finite MTL-algebra. Then (I _α  (M) , ∩ , ∨  _α , {0} , M) is a Boolean algebra.

Definition 5.13 ([6]) Assume M is a complete lattice and δ ∈ M. If for any G ⊆ M, δ ≤ ⋁ G implies that there exist ξ₁ ·· · ξ _n  ∈ G such that δ ≤ ξ₁ ∨ ·· · ∨ ξ _n , then we say a is a compact element. And we say M is a algebraic lattice provided that its each element can be expressed as the join of some compact elements.

Theorem 5.14. Let ξ ∈ M, then the least α-ideal containing 〈ξ〉 is compact in I _α  (M), i.e. ⋃ δ ∈ 〈 ξ 〉 δ ⊤ ⊤ is compact in I _α  (M).

Proof. From the proof of Theorem 5.9, we can see that ⋃ δ ∈ 〈 ξ 〉 δ ⊤ ⊤ ∈ I α ( M ) . Suppose A ⊆ I α ( M ) such that ⋃ δ ∈ 〈 ξ 〉 δ ⊤ ⊤ ⊆ ⋁ α A , we aim to show there exist finite elements G₁, G₂ ·· · G _n of A satisfying ⋃ δ ∈ 〈 ξ 〉 δ ⊤ ⊤ ⊆ G 1 ∨ α G 2 ∨ α · · · ∨ α G n .

Since ξ ∈ 〈ξ〉 and ξ ∈ ξ^⊤⊤, we have ξ ∈ ⋁ α A = ⋃ θ ∈ ⋁ I A θ ⊤ ⊤ . So there is an θ ∈ ⋁ I A such that ξ^⊤⊤ ⊆ θ^⊤⊤. In fact, by Theorem 3.5 θ ∈ ⋁ I A implies there exist G 1 , G 2 · · · G n ⊆ A and δ _i  ∈ G _i (1 ≤ i ≤ n) such that θ ≤ δ _n  ⊕ (⋯ ⊕ (δ₃ ⊕ (δ₂ ⊕ δ₁)) ⋯). It follows from Proposition 4.6 (i) that θ ⊤ ⊤ ⊆ ( δ n ⊕ ( ⋯ ⊕ ( δ 3 ⊕ ( δ 2 ⊕ δ 1 ) ) ⋯ ) ) ⊤ ⊤ . So we can get ξ ∈ ξ ⊤ ⊤ ⊆ ( δ n ⊕ ( ⋯ ⊕ ( δ 3 ⊕ ( δ 2 ⊕ δ 1 ) ) ⋯ ) ) ⊤ ⊤ . Let B = { G 1 , G 2 · · · G n } , it is easy to see that δ n ⊕ ( ⋯ ⊕ ( δ 3 ⊕ ( δ 2 ⊕ δ 1 ) ) ⋯ ) ∈ ⋁ α B , which means ξ ∈ ⋁ α B . In fact, ⋁ α B is an α-ideal, it gives 〈 ξ 〉 ⊆ ⋁ α B . Since ⋃ δ ∈ 〈 ξ 〉 δ ⊤ ⊤ is the least α-ideal containing 〈ξ〉, we have ⋃ δ ∈ 〈 ξ 〉 δ ⊤ ⊤ ⊆ ⋁ α B . By the fact that B is finite, we conclude that ⋃ δ ∈ 〈 ξ 〉 δ ⊤ ⊤ is compact.

Conversely, we state that each compact element in I _α  (M) is with the form ⋃ u ∈ 〈 z 〉 u ⊤ ⊤ , for some z ∈ M. Let G ∈ I _α  (M) be a compact element of I _α  (M). Then by Theorem 5.5 we get G = ⋃ ξ ∈ G ξ ⊤ ⊤ . Since G = ⋁  _I  {〈δ〉|δ ∈ G}, we have

G = ⋃ ξ ∈ ⋁ I { 〈 δ 〉 | δ ∈ G } ξ ⊤ ⊤ = ⋁ α { ⋃ ξ ∈ 〈 δ 〉 ξ ⊤ ⊤ | δ ∈ G } . Since G is compact, there exist δ _i  ∈ G (1 ≤ i ≤ n) such that G = ( ⋃ ξ ∈ 〈 δ 1 〉 ξ ⊤ ⊤ ) ⋁ α · · · ⋁ α ( ⋃ ξ ∈ 〈 δ n 〉 ξ ⊤ ⊤ ) = ⋃ ξ ∈ 〈 δ 1 〉 ∨ I · · · ∨ I 〈 δ n 〉 ξ ⊤ ⊤ . It follows from Proposition 3.7(ii) that G = ⋃ ξ ∈ 〈 δ 1 ⊕ · · · ⊕ δ n 〉 ξ ⊤ ⊤ . Put z = δ₁ ⊕ ·· · ⊕ δ _n , we have G = ⋃ ξ ∈ 〈 z 〉 ξ ⊤ ⊤ .

□

Theorem 5.15. The lattice (I _α  (M) , ∩ , ∨  _α , {0} , M) is an algebraic lattice.

Proof. Let G ∈ I _α  (M). Then by Theorem 5.5 we get G = ⋃ ξ ∈ G ξ ⊤ ⊤ . Since G = ⋁  _I  {〈δ〉|δ ∈ G}, we have

G = ⋃ ξ ∈ ⋁ I { 〈 δ 〉 | δ ∈ G } ξ ⊤ ⊤ = ⋁ α { ⋃ ξ ∈ 〈 δ 〉 ξ ⊤ ⊤ | δ ∈ G } . It follows from Theorem 5.14 that ⋃ ξ ∈ 〈 δ 〉 ξ ⊤ ⊤ is compact. Therefore, (I _α  (M) , ∩ , ∨  _α , {0} , M) is an algebraic lattice. □

Theorem 5.16. The lattice structure (I _α  (M) , ∩ , ∨  _α , {0} , M) can form a Heyting algebra.

Proof. For any I, J ∈ I _α  (M), consider U = {x ∈ M| ∀ i ∈ I, i ∧ x ∈ J}. Now, we prove U is an ideal. If a, b ∈ U, then for any i ∈ I, a ∧ i ∈ J, b ∧ i ∈ J, so by Proposition 3.2(vii) (a ⊕ b) ∧ i ≤ (a ∧ i) ⊕ (b ∧ i) ∈ J. If a ∈ U and b ≤ a, then for any i ∈ I, b ∧ i ≤ a ∧ i ∈ J, thus b ∧ i ∈ J, it implies b ∈ U. Therefore, U is an ideal. Define I →  _α J = ⋃ _z∈Uz^⊤⊤. Clearly, I →  _α J ∈ I _α  (M). Next, we prove H ∩ I ⊆ J iff H ⊆ I →  _α J   If I, J, H ∈ I _α  (M) and H ∩ I ⊆ J. We will show that H ⊆ I →  _α J. Let x ∈ H, it is easy to see that x ∧ i ∈ H ∩ I, so x ∧ i ∈ J for all i ∈ I. Hence x ∈ I →  _α J. Since x is arbitrary, we conclude that H ⊆ I →  _α J.

  Conversely, assume H ⊆ I →  _α J. Now we prove that H ∩ I ⊆ J. Let y ∈ H ∩ I, then y ∈ I and y ∈ H, and so y ∈ I →  _α J = ⋃ _z∈Uz^⊤⊤. Then there exists z ∈ U such that y ∈ z^⊤⊤. Further, we get y^⊤⊤ ⊆ z^⊤⊤ and z ∧ i ∈ J for all i ∈ I. Put i = y, then y ∧ z ∈ J. Since J ∈ I _α  (M), we have (y ∧ z) ^⊤⊤ ⊆ J. By Proposition 4.6(iv), we get (y ∧ z) ^⊤⊤ = y^⊤⊤ ∩ z^⊤⊤ = y^⊤⊤. It yields that y ∈ y^⊤⊤ ⊆ J. Since y is arbitrary, we conclude that H ∩ I ⊆ J. □

6 Conclusions

In this paper, we showed that ⊕ is non-associative on residuated lattices. It follows that the ideal generation formula in BL-algebras [11] doesn’t hold on residuated lattices. So it is clearly incorrect that Liu [12] extended the ideal generation formula in BL-algebras to residuated lattices. But in our paper, we used the inequality (δ ⊕ ξ) ⊕ θ ≤ δ ⊕ (ξ ⊕ θ) to give the correct ideal generation formula on residuated lattices. It laid the foundation for us to study ideals on residuated lattices. Also, we extended the concepts of annihilators and α-ideals to MTL-algebras, and generalized some results obtained in [24]. We showed that the least α-ideal containing the ideal I is ⋃_x∈Ix^⊤⊤ on an MTL-algebra and we further proved that the least α-ideal containing the ideal I is I^⊤⊤ on a finite MTL-algebra. In fact, we observed that there are no non-trivial α-ideals on MTL-chains or in BL-chains. Thus, the set I _α  (M) of all α-ideals in a BL-chain M [24] was a trivial structure containing two elements {0} and M. However, Zou [24] discussed the lattice structure of I _α  (M) in a BL-chain M and obtained some related results. Obviously, these results were trivial in [24]. While in our paper, we studied the lattice structure of I _α  (M) on a general MTL-algebra M and got a series of non-trivial results in some non-chain cases. For example, I _α  (M) can be made into an algebraic lattice, (I _α  (M) , ∩ , ∨  _α , {0} , M) became a Boolean algebra under certain conditions. Future research will focus on studying the congruence relation induced by α-ideals and corresponding quotient algebra on MTL-algebras.