Study on the properties of A-subset

1 Introduction

Non-classical logic has become a considerable formal tool for computer science and artificial intelligence to deal with fuzzy information and uncertain information. Many-valued logic, a extension and development of classical logic [1], has always been a crucial direction in non-classical logic. Incomparability which can be encountered in our life is an important one among all kinds of uncertainties. In order to research the many-valued logical system whose propositional value is given in a lattice, in 1993 Xu [2, 3] proposed the concept of lattice implication algebras by combining algebraic lattice and implication algebra. Since then many researchers have investigated this logic algebra and many results have been obtained [9–13, 24–29]. For example, in [4, 5], Jun et al. introduced the concept of LI-ideals and prime LI-ideals in lattice implication algebras and investigated some properties. In [6, 7], Liu introduced the notions of ILI-ideals and maximal LI-ideals in lattice implication algebras and investigated the properties of ILI-ideals, prime LI-ideals and maximal LI-ideals, respectively. In [8], Zhao studied on the implicative and fuzzy implicative LI-ideals in lattice implication algebras. In [20], Long investigated the LI-ideals in the lattice implication product algebras L₁ × L₂. After that, Long studied on fuzzy LI-ideals in the lattice implication product algebras and discussed the relation between the fuzzy LI-ideals of L₁ × L₂ and the fuzzy LI-ideals of L₁ and L₂ [21]. In [23], Zhu proposed the concept of IFI–ideals in lattice implication algebras and discussed the related properties. Xu have collected details of lattice implication algebras and lattice-value logic based on lattice implicationalgebras [3].

In 2005, Zhu proposed the concept of the annihilator in lattice implication algebras and introduced related properties [14]. The annihilator is an important part of the algebraic structure in lattice implication algebras. Since then, Long studied the properties of annihilator carefully in lattice implication algebras. In [15], Long proposed the concept of fuzzy annihilator in lattice implication algebras and some properties. In [19], Long intensive studied the properties of annihilators in finite lattice implication algebras. In this article, the authors founded out all annihilators of LI-ideals in the finite lattice implication algebras and proved that doing annihilator (briefly 0^*) was an order-reversing involution operator. Next, the authors defined the implicative operator “⇒” in the Σ (L)(the sets of all LI-ideals of L) and proved that (Σ (L) ,   ∨ ,   ∧ ,   ⇒ ,  0^*,   {0} ,  L) was a lattice implication algebra.

In 2006, Long proposed the concept of α-subsets which it was important promotion of annihilators [16]. The notion of α-subsets was gained that the zero element was replaced by a nonzero element in lattice implication algebras. Next, Long investigated the related properties of α-subsets [16–18]. It is very close that the relations between α-subsets and α-resolution domain. The proposition of α-subsets provided some help for automatic reasoning.

Now, a natural idea is to replace an element with a set. In this paper, with this objective in view, we will propose the concept of A-subsets and discuss the properties in lattice implication algebras. In particular, we will focus on the properties of A-subset when A is an LI-ideal and prove that B (A)(the A-subset of B) is an LI-ideal and A ⊆ B (A) in lattice implication algebra. Next, we will study on the A-subset in lattice implication product algebras. Firstly, we investigate the properties of A-subset in lattice implication product algebra L₁ × L₂. It is obtained that the A-subset of B₁ × B₂ is the product of A-subset of L _i  (i = 1,  2) respectively (briefly(B₁ × B₂) (A₁ × A₂) = B₁ (A₁) × B₂ (A₂)). Finally, we will generalize the conclusion from the product of two lattice implication algebras to the product of many lattice implication algebras. It is hoped that the above work would serve as a foundation for further studying lattice-valued logic system based on linguistic truth values lattice implication algebras and developing corresponding applied methods of uncertainty reasoning and resolution automated reasoning with linguisticterms.

2 Preliminaries

In this section, we recall some basic concepts and results, which we shall need in the subsequentsections.

Lattice implication algebras, introduced by Xu, are important logic algebras. A lattice implicationalgebra is defined as follows:

Definition 2.1. (Xu [2]) Let (L,   ∨ ,   ∧ , ^′,  0,  I) be a bounded lattice with the universal boundaries 0 (the least element) and I(the greatest element) respectively, and “^′” an order-reversing involution. If a mapping → : L × L → L satisfies for anyx,  y,  z ∈ L,

x → (y → z) = y → (x → z);

A mapping f : L₁ → L₂ from lattice implication algebras L₁ to L₂ is called a lattice implication homomorphism if it satisfies: for any x,  y ∈ L₁ f ( x → y ) = f ( x ) → f ( y ) f ( x ∨ y ) = f ( x ) ∨ f ( y ) f ( x ∧ y ) = f ( x ) ∧ f ( y ) f ( x ′ ) = ( f ( x ) ) ′

A one-to-one and onto lattice implication homomorphism is called a lattice implication isomorphism.

In a lattice implication algebra, we define a partial ordering “⩽” as follows: ( ∀ x , y ∈ L ) ( x ⩽ y ⇔ x → y = I ) and the operation “⊕” as follows: ( ∀ x , y ∈ L ) ( x ⊕ y = x ′ → y )

In a lattice implication algebra L, the following hold:

Theorem 2.1.(Xu [2]) LetLbe a lattice implication algebra, for anyx,  y, z ∈ Lthe following hold:

0 → x = I,  x → 0 = x′, I → x = x,  x → I = I;

For more details of lattice implication algebras we refer the readers to [3].

Definition 2.2. (Xu [2]) Let L be a lattice implication algebra. A non-empty subset A of L is called an LI-ideal of L if it satisfies the following conditions:

(LI1) 0 ∈ A;

(LI2) (∀ x,  y ∈ L) ((x → y) ′ ∈ A, y ∈ A ⇒ x ∈ A).

Let ∑ (L) denote the set of all LI-ideals of L.

Theorem 2.2.(Xu [2]) LetLbe a lattice implication algebra.Anon-empty subsetAofLis called an LI-ideal ofLif and only if:

(LI3) (∀ x,  y ∈ L) (x ⩽ y,  y ∈ A ⇒ x ∈ A);

(LI4) (∀ x,  y ∈ L) ((x ∈ A,  y ∈ A ⇒ x ⊕ y ∈ A).

Theorem 2.3.(Liu, Xu and Qin [6]) LetLbe a lattice implication algebra. A non-empty subsetAofLis called an LI-ideal ofL ⇔ (∀ x,  y ∈ A,   ∀ z ∈ L) if (z → x) ′ ⩽ y, then z ∈ A.

Definition 2.3. (Jun, Roh, and Xu [4]) Let L be a lattice implication algebra. A is a non-empty subset of L, the set <A>that it is generated by A is called an LI-ideal of L and a minimal LI-ideal contained A. Especially, if A = {a} ,  < A > = < {a} > = < a >;.

Theorem 2.4.(Jun, Roh, and Xu [4]) LetLbe a lattice implication algebra. A is a non-empty subset ofL, the following hold: ( 1 ) < A > = { x ∈ L | ∃ a 1 , a 2 , ⋯ , a n ∈ A , n ∈ N * , s . t . a 1 ′ → ( ⋯ → ( a n ′ → x ) ⋯ ) = I } ; ( 2 ) < A > = { x ∈ L | ∃ a 1 , a 2 , ⋯ , a n ∈ A , n ∈ N * , s . t . x ⩽ a 1 ⊕ a 2 ⊕ ⋯ ⊕ a n } ; ( 3 ) < a > = { x ∈ L | ∃ n ∈ N * , s . t . x ⩽ na } .

Theorem 2.5.(Xu [2]) LetLbe a lattice implication algebra, the following hold: ( 1 ) ( ∀ A , B ∈ Σ ( L ) ) ( A ⊆ B ⇒ < A > ⊆ < B > ) ; ( 2 ) ( ∀ x , y ∈ L ) ( x ⩽ y ⇒ < x > ⊆ < y > ) ; ( 3 ) ( ∀ x ∈ L ) ( ∀ A ∈ Σ ( L ) ) ( < x > ⊆ A ⇔ x ∈ A ) ; ( 4 ) ( ∀ A , B ∈ Σ ( L ) ) ( < A ∪ B > = { x ∈ L | ∃ a ∈ A , b ∈ B , s . t . x ⩽ a ⊕ b } ) Theorem 2.6.(Jun, Roh, and Xu [4]) LetLbe a lattice implication algebra, the following hold: ( 1 ) ( ∀ a , b ∈ L ) ( < a > ∩ < b > = < a ∧ b > ) ; ( 2 ) ( ∀ a , b ∈ L ) ( < < a > ∪ < b > > = < a ∨ b > ) .

Definition 2.4. (Jun, Roh, and Xu [4]) Let L be a lattice implication algebra. A proper LI-ideal A of a lattice implication algebra L is said to be prime if it satisfies the following condition: ( ∀ x , y ∈ L ) ( x ∧ y ∈ A ⇒ x ∈ A or y ∈ A ) ; A proper LI-ideal A of a lattice implication algebra L is said to be maximal if it satisfies the following condition: ( ∀ B ( B ≠ L ) ∈ ∑ ( L ) ) ( A ⊄ B )

Definition 2.5. (Liu, Xu and Qin [6]) Let L be alattice implication algebra. An LI-ideal A of a lattice implication algebra L is said to be an ILI-ideal if it satisfies the following condition: ( ∀ x , y , z ∈ L ) ( ( If z ∈ A and ( ( ( x → y ) ′ → y ) ′ → z ) ′ ∈ A ⇒ ( x → y ) ′ ∈ A

Theorem 2.7.(Liu, Xu and Qin [6]) LetLbe a lattice implication algebra, ∀A,  B ∈ ∑ (L), ifA ⊆ BandAis an ILI-ideal ofL, thenBis an ILI-ideal ofL.

3 A-subset

Firstly, the notion of the A-subset is introduced in lattice implication algebras.

Definition 3. 1. Let L be a lattice implication algebra. A and B are two non-empty subsets of L,

B (A) ={x ∈ L|x ∧ b ∈ A,   ∀ b ∈ B}

B (A) is called an A-subset of B.

Obviously, if A = {0}, B (A) = B (0) is an annihilator of B.

Example 3.1. Let L = {0,  a,  b,  c,  d,  I} the Hasse diagram of L be defined as Fig. 1 and “^′” and “→” as follows:

OPEN IN VIEWER

It is easy to verify that (L,   ∧ ,   ∨ , ′,   →) is a lattice implication algebra [6]. Let B = {a,  b} ,  A₁ = {0,  c}, A₂ = {0,  d,  a} ,  A₃ = {c,  d}, then B (A₁) = {0,  c}, B (A₂) = {0,  d,  a} ,  B (A₃) = {d}.

The example 3.1 shows that the A-subset exists in lattice implication algebras.

Next, the properties of A-subset are discussed in lattice implication algebras.

Remark 3.1.B (A) ² = (B (A)) (A), B (A) ³ = ((B (A)) (A)) (A).

Theorem 3.1.Let L be a lattice implication algebra, Ais a non-empty subset ofL, the following hold:

(∀ B ⊆ C ⊆ L) (C (A) ⊆ B (A));

Proof. (1) By Definition 3.1, suppose x ∈ C (A), then ∀c ∈ C,  x ∧ c ∈ A, because B ⊆ C, ∀b ∈ B ⇒ b ∈ C, x ∧ b ∈ A, then x ∈ B (A), hence C (A) ⊆ B (A).

(2) Suppose x ∈ B, for any y ∈ B (A), ∀b ∈ B, y ∧ b ∈ A, so y ∧ x ∈ A, by Definition 3.1,x ∈ B (A) ², hence B ⊆ B (A) ².

(3) By the Theorem 3.1 (1) (2), because B ⊆ B (A) ², then B (A) ³ ⊆ B (A). On the other hand, suppose x ∈ B (A), ∀c ∈ B (A) ², ∀d ∈ B (A), c ∧ d ∈ A, then x ∧ c ∈ A, x ∈ B (A) ³, hence B (A) ⊆ B (A) ³. So, B (A) = B (A) ³.

(4) Because B ⊆ B ∪ C,  C ⊆ B ∪ C, by the Theorem 3.1 (1), (B ∪ C) (A) ⊆ B (A) , (B ∪ C) (A) ⊆ C (A) hence (B ∪ C) (A) ⊆ B (A) ∩ C (A). □

Theorem 3.2.LetLbe a lattice implication algebra, Cis a non-empty subset ofL, the following hold:

(∀ A ⊆ B ⊆ L) (C (A) ⊆ C (B));

Proof. (1) By Definition 3.1, suppose x ∈ C (A), then ∀c ∈ C,  x ∧ c ∈ A, because A ⊆ B, then x ∧ c ∈ B, x ∈ C (B), hence C (A) ⊆ C (B).

(2) Suppose ∀x ∈ (C (A) ∪ C (B)), then x ∈ C (A) or x ∈ C (B), ∀c ∈ C, x ∧ c ∈ A or x ∧ c ∈ B, hence x ∧ c ∈ A ∪ B, x ∈ C (A ∪ B), thus C (A) ∪ C (B) ⊆ C (A ∪ B); On the other hand, suppose x ∈ C (A ∪ B), ∀c ∈ C, x ∧ c ∈ A ∪ B, then x ∧ c ∈ A or x ∧ c ∈ B, x ∈ C (A) or x ∈ C (B), hence x ∈ (C (A) ∪ C (B)), thus C (A ∪ B) ⊆ C (A) ∪ C (B), so C (A) ∪ C (B) = C (A ∪ B).

(3) Because A ∩ B ⊆ A,  A ∩ B ⊆ B, by the Theorem 3.1 (1), C (A ∩ B) ⊆ C (A), C (A ∩ B) ⊆ C (B), then C (A ∩ B) ⊆ C (A) ∩ C (B); On the other hand, suppose x ∈ C (A) ∩ C (B), then x ∈ C (A) and x ∈ C (B), ∀c ∈ C, x ∧ c ∈ A and x ∧ c ∈ B, so x ∧ c ∈ A ∩ B, x ∈ C (A ∩ B), then C (A) ∩ C (B) ⊆ C (A ∩ B), hence C (A) ∩ C (B) = C (A ∩ B). □

Moreover, the properties of A-subset are introduced in lattice implication algebras when A is an LI-ideal. It is obtained that B (A) is an LI-ideal of L and A ⊆ B (A). These conclusions are very important for us to further study the structure of the lattice implication algebra.

Theorem 3.3.LetLbe a lattice implication algebra, Bis a non-empty subset ofL, Ais an LI-ideal ofL, thenB (A) is an LI-ideal ofLandA ⊆ B (A).

Proof. ∀b ∈ B,  0 ∧ b = 0 ∈ A, then 0 ∈ B (A); ∀x,  y ∈ L, if (x → y) ′ ∈ B (A) and y ∈ B (A), then ∀b ∈ B, (x → y) ′ ∧ b ∈ A and y ∧ b ∈ A; A is an LI-ideal of L, by the Theorem 2.2, (y ∧ b) ⊕ ((x → y) ′ ∧ b) ∈ A, by the Theorem 2.1 (5) (8), (y ∧ b) ⊕ ((x → y) ′ ∧ b) ⩾ b ∧ (y ⊕ (x → y) ′) = b ∧ (x ∨ y) ⩾ b ∧ x, so x ∧ b ∈ A, x ∈ B (A), then B (A) is an LI-ideal of L; Next, we will prove the A ⊆ B (A). If x ∈ A, ∀b ∈ B,  x ∧ b ⩽ x, by A is an LI-ideal of L, x ∧ b ∈ A, then x ∈ B (A), A ⊆ B (A). □

Theorem 3.4.LetLbe a lattice implication algebra, Bis a non-empty subset ofL, Ais an LI-ideal ofL, thenB (A) = L ⇔ B ⊆ A.

Proof. “⇐” Obviously, B (A) ⊆ L. Next, we will prove the L ⊆ B (A), suppose x ∈ L, because B ⊆ A, ∀b ∈ B, then b ∈ A, x ∧ b ⩽ b ∈ A, A is an LI-ideal of L, so x ∧ b ∈ A, then x ∈ B (A), L ⊆ B (A), hence B (A) = L.

“⇒” If B (A) = L, ∀b ∈ B, then b ∈ B ⊆ L = B (A), so b ∧ b = b ∈ A, thus B ⊆ A. □

Theorem 3.5.LetLbe a lattice implication algebra, the following hold:

( 1 ) ( ∀ A ∈ ∑ ( L ) , B ⊆ L ) ( A ( A ) = 0 ( A ) = ( B ( A ) ∩ B ) ( A ) = L ) ; ( 2 ) ( ∀ A , B ∈ ∑ ( L ) ) ( A ⊆ B ⇒ B ( A ) ∩ B ) = A ) ; ( 3 ) ( ∀ A ∈ ∑ ( L ) , B ⊆ L ) ( B ( A ) 2 ∩ B ( A ) = A ) ; ( 4 ) ( ∀ A ∈ ∑ ( L ) , B ⊆ L ) ( B ( A ) = < B > ( A ) ) ; ( 5 ) ( ∀ A ∈ ∑ ( L ) , B , C ⊆ L ) ( C ( B ( A ) ) = B ( C ( A ) ) ) ; ( 6 ) ( ∀ A ∈ ∑ ( L ) , B ⊆ L ) ( A ⊆ B ⇒ < B > ∩ < B > ( A ) = A ) , Where 0 ( A ) = { 0 } ( A ) .

Proof. (1) Suppose x ∈ B (A) ∩ B, then x ∈ B (A) and x ∈ B, by Definition 3.1, x ∧ x = x ∈ A, thus B (A) ∩ B ⊆ A; In addition, {0} ⊆ A,  A ⊆ A, by Theorem 3.4, 0 (A) ⊆ A (A) = (B (A) ∩ B) (A) = L.

(2) Suppose A ⊆ B, by Theorem 4.1, A ⊆ B (A), so A ⊆ B (A) ∩ B; On the other hand, suppose x ∈ B (A) ∩ B, then x ∈ B (A) and x ∈ B, by Definition 3.1, x ∧ x = x ∈ A, thus B (A) ∩ B ⊆ A, hence B (A) ∩ B) = A.

(3) By Theorem 4.1 and Theorem 3.2 (3), B (A) ² ∩ B (A) = A.

(4) Because B⊆ < B >, by the Theorem 3.1 (1) <B > (A) ⊆ B (A); On the other hand, suppose x ∈ B (A), ∀b ∈ B,  x ∧ b ∈ A, ∀b₁ ∈ < B > , then ∃a₁,  a₂, ⋯ ,  a _n  ∈ B, s.t. b₁ ⩽ a₁ ⊕ a₂ ⊕ ⋯ ⊕ a _n , by Theorem 2.1 (8), b₁ ∧ x = x ∧ b₁ ⩽ x ∧ (a₁ ⊕ a₂ ⊕ ⋯ ⊕ a _n ) ⩽ (x ∧ a₁) ⊕ (x ∧ a₂) ⊕ ⋯ ⊕ (x ∧ a _n ), because x ∧ a _i  ∈ A (i = 1,  2, ⋯ ,  n), A is an LI-ideal of L, by Theorem 2.4, x ∧ b₁ ∈ A, so x ∈ < B > (A), then B (A) ⊆ < B > (A), hence B (A) = < B > (A).

(5) Suppose x ∈ C (B (A)), ∀c ∈ C,  x ∧ c ∈ B (A), by Definition 3.1, ∀b ∈ B, (x ∧ c) ∧ b ∈ A, then (x ∧ b) ∧ c ∈ A, (x ∧ b) ∈ C (A), x ∈ B (C (A)), so C (B (A)) ⊆ B (C (A)); In the same way, we can prove that B (C (A)) ⊆ C (B (A)), hence C (B (A)) = B (C (A)).

(6) Because A ⊆ B, A⊆ B ⊆ < B >, by Theorem 3.5 (2), <B > ∩ < B > (A) = A. □

Theorem 3.6.LetLbe a lattice implication algebra, Ais an LI-ideal ofL, B,  Care two non-empty subsets ofL, the following hold:

If B = ∅ ⇒ B (A) = A;

Proof. (1) Trivially.

(2) Suppose I ∈ B, ∀x ∈ B (A) ,  x = x ∧ I ∈ A, then B (A) ⊆ A; By the Theorem 3.3, A ⊆ B (A), hence B (A) = A.

(3) If B (A) ≠ A, then ∃x ∈ B (A) ,  s . t .  x ∉ A, ∀b ∈ B, x ∧ b ∈ A, because A is a prime LI-ideal of L, then b ∈ A, B ⊆ A, By the Theorem 3.4, B (A) = L; Contradiction, so B (A) = A.

(4) If A is a maximal LI-ideal of L, B (A) ≠ L, then B (A) ⊆ A; By Theorem 3.3, A ⊆ B (A), hence B (A) = A.

(5) Suppose B ⊆ C,  B (A) = A, by the Theorem 3.1 (1), C (A) ⊆ B (A); By Theorem 3.3, A ⊆ C (A), hence C (A) = A. □

Theorem 3.7.LetLbe a lattice implication algebra, Ais an LI-ideal ofL, Bis a non-empty subset ofL, if A is an ILI-ideal ofL, B (A) is an ILI-ideal ofL.

Proof. Because A is an ILI-ideal of L, by the Theorem3.3, A ⊆ B (A), hence by the Theorem 2.7, B (A) is an ILI-ideal of L. □

Theorem 3.8.LetLbe a lattice implication algebra, {A_α } _α∈J ⊆ ∑ (L) andB ⊆ L, then ∩_α∈JB (A_α) = B (∩ _α∈JA_α), Jis an index set.

Proof. For any x ∈ ∩ _α∈JB (A_α) ⇔ (∀ α ∈ J) (x ∈ B (A_α)) ⇔ (∀ α ∈ J) (∀ b ∈ B)(x ∧ b ∈ A_α) ⇔ (∀ b ∈ B) (x ∧ b ∈ ∩ _α∈JA_α) ⇔ x ∈ B (∩ _α∈JA_α)), then ∩_α∈JB (A_α) = B (∩ _α∈JA_α). □

Remark 3.2.Ψ (B) = { B (A) | ∀ A ∈ ∑ (L)  } (∀ B ⊆ L).

In the following, we will prove that Ψ (B) is closed under the operation ∩.

Theorem 3.9.LetLbe a lattice implication algebra, Ψ (B) is closed under the operation ∩.

Proof. For any B (A₁) ,  B (A₂) ∈ Ψ (B), A₁,  A₂ ∈ Σ (L), then A₁ ∩ A₂ ∈ Σ (L), by the Theorem 3.8, B (A₁) ∩ B (A₂) = B (A₁ ∩ A₂) ∈ Ψ (B). □

Next, the propositions of homomorphism image of A-subset are investigated in the following.

Theorem 3.10.LetL₁andL₂be two lattice implication algebras, the mappingf : L₁ → L₂is a lattice implication homomorphism, ∀B ⊆ L₁,  A ∈ Σ (L₁), thenf (B (A)) ⊆ (f (B)) (f (A)).

Proof. For any y ∈ f (B (A)) ,    ∃ x ∈ B (A) , s . t . f ( x ) = y , because x ∈ B (A) ,   ∀ b ∈ B, x ∧ b ∈ A, then f (b) ∈ f (B) ,   f (x ∧ b) = f (x) ∧ f (b) ∈ f (A), hence y = f (x) ∈ (f (B)) (f (A)). □

Corollary 3.11.LetL₁andL₂be two lattice implication algebras, the mappingf : L₁ → L₂is a lattice implication isomorphism, ∀B ⊆ L₁,  A ∈ Σ (L₁), thenf (B (A)) = (f (B)) (f (A)).

Proof. It can be obtained by Theorem 3.10 and the definition of the lattice implication isomorphism. □

4 A-subset of the lattice implication product algebra

In this section, we focus on the A-subset in lattice implication product algebras. Firstly, we investigate the properties of A-subset in lattice implication product algebra L₁ × L₂. It is obtained that the A-subset of B₁ × B₂ is the product of A-subsets of L _i  (i = 1,  2) respectively ((B₁ × B₂) (A₁ × A₂) = B₁ (A₁) × B₂ (A₂)). In the end, we will generalize the conclusion from the product of two lattice implication algebras to the product of many lattice implication algebras.

Definition 4.1. (Xu, Ruan, Qin and Liu [3]) Let ( L α , ∨ α , ∧ α , ( α ) ′ , → α , 0 α , I α ) ( α ∈ J ) be a family of lattice implication algebras, where J is an index set. Define L = ∏ { L α | α ∈ J } = { f | f : J → ⋃ α ∈ J L α , s . t . for any α ∈ J , f ( α ) ∈ L α } .

For any f,  g ∈ A,  α ∈ J, binary operation ∨,  ∧ ,   → and an unary operation on L are defined as follows: ( f ∨ g ) ( α ) = f ( α ) ∨ α g ( α ) ( f ∧ g ) ( α ) = f ( α ) ∧ α g ( α ) f ′ ( α ) = ( f ( α ) ) ′ ( α ) ( f → g ) ( α ) = f ( α ) → α g ( α )

and 0 (α) =0_α,  I (α) = I_α. It can be proved that these operations are well defined and L is a lattice implication algebra. It is called the direct product or a lattice implication product algebra of L_α (α ∈ J). Specially, if J ={ 1,  2 }, then L = L₁ × L₂ is a lattice implication product algebra of L₁ and L₂.

Theorem 4.1. (Long [20]) LetL₁andL₂be two lattice implication algebras, L₁ × L₂their lattice implication product algebra. For anyA _i  ⊆ L _i  (i = 1,  2), A₁ × A₂is an LI-ideal (or a prime LI-ideal) of the lattice implication product algebraL₁ × L₂if and only ifA _i  (i = 1,  2) is an LI-ideal (or a prime LI-ideal) of the lattice implication algebraL _i  (i = 1,  2) respectively.

Theorem 4.2.LetL₁andL₂be two lattice implication algebras, L₁ × L₂their lattice implication product algebra. For anyA _i  ∈ ∑ (L _i ) andB _i  ⊆ L _i  (i = 1,  2), then (B₁ × B₂) (A₁ × A₂) = B₁ (A₁) × B₂ (A₂) .

Proof. For any x = (x₁,  x₂) ∈ (B₁ × B₂) (A₁ × A₂), then ∀b = (b₁,  b₂) ∈ B₁ × B₂, x∧ b ∈ A₁ × A₂, hence, (x₁ ∧ b₁) × (x₂ ∧ b₂) ∈ A₁ × A₂, x₁ ∧ b₁ ∈ A₁, x₂ ∧ b₂ ∈ A₂, so x₁ ∈ B₁ (A₁), x₂ ∈ B₂ (A₂), x ∈ B₁ (A₁) × B₂ (A₂).

On the other hand, for any y = (y₁,  y₂) ∈ B₁ (A₁) × B₂ (A₂), then y₁ ∈ B₁ (A₁) and y₂ ∈ B₂ (A₂), hence ∀b₁ ∈ B₁,  y₁ ∧ b₁ ∈ A₁ and ∀b₂ ∈ B₂,  y₂ ∧ b₂ ∈ A₂, note b = (b₁,  b₂) ∈ B₁ × B₂, y ∧ b = (y₁ ∧ b₁) × (y₂ ∧ b₂) ∈ A₁ × A₂, so y ∈ (B₁ × B₂) (A₁ × A₂). Consequently, (B₁ × B₂) (A₁ × A₂) = B₁ (A₁) × B₂ (A₂) . □

Theorem 4.3.LetL₁andL₂be two lattice implication algebras, L₁ × L₂their lattice implication product algebra. For anyA _i is a prime Li-ideal ofL _i , B _i  ⊆ L _i andB _i  (A _i ) ≠ L _i  (i = 1, 2), then ( B 1 × B 2 ) ( A 1 × A 2 ) = B 1 ( A 1 ) × B 2 ( A 2 ) = A 1 × A 2 .

Proof. It can be obtained by the Theorem 3.6 (3) and the Theorem 4.2. □

Next, we will generalize the conclusion from the product of two lattice implication algebras to the product of many lattice implication algebras.

Theorem 4.4.LetL _i  (i = 1,  2, ⋯ ,  n) be lattice implication algebras, ∏ i = 1 n L i their lattice implication product algebra, 0 _i  (i = 1,  2, ⋯ ,  n) is the smallest element ofL _i  (i = 1,  2, ⋯ ,  n). For anyA _i  ⊆ L _i  (i = 1,  2, ⋯ ,  n), ∏ i = 1 n A i is an LI-ideal of the lattice implication product algebra ∏ i = 1 n L i if and only ifA _i  (i = 1,  2, ⋯ ,  n) is an LI-ideal of the lattice implication algebraL _i  (i = 1,  2, ⋯ ,  n) respectively.

Proof. “⇒” Because ∏ i = 1 n A i is an LI-ideal of the lattice implication product algebra ∏ i = 1 n L i , then ( 0 1 , 0 2 , ⋯ , 0 n ) ∈ ∏ i = 1 n A i , 0 _i  ∈ A _i  (i = 1,  2, ⋯ ,  n). For any x _i ,  y _i  ∈ A _i  (i = 1,  2, ⋯ ,  n), x = ( x 1 , x 2 , ⋯ , x n ) ∈ ∏ i = 1 n A i , y = ( y 1 , y 2 , ⋯ , y n ) ∈ ∏ i = 1 n A i , ∀z _i  ∈ L _i  (i = 1,  2, ⋯ ,  n), then z = ( z 1 , z 2 , ⋯ , z n ) ∈ ∏ i = 1 n L i . If ∀i = {1, 2, ⋯ , n} ,   (z _i  → x _i ) ′ ⩽ y _i , hence ((z₁ → x₁) ′, (z₂ → x₂) ′, ⋯ ,   (z _n  → x _n ) ′) = (z₁ → x₁, z₂ → x₂, ⋯ ,  z _n  → x _n ) ′ = ((z₁, z₂, ⋯ , z _n ) → (x₁, x₂, ⋯, x _n )) ′⩽ (y₁,  y₂, ⋯ ,  y _n ), then (z → x) ′ ⩽ y, by the Theorem 2.3, we have z = ( z 1 , z 2 , ⋯ , z n ) ∈ ∏ i = 1 n A i and z _i  ∈ A _i  (i = 1,  2, ⋯ ,  n), so A _i  (i = 1,  2, ⋯ ,  n) is an LI-ideal of the lattice implication algebra L _i  (i = 1,  2, ⋯ ,  n) respectively.

“⇐” Because A _i  (i = 1,  2, ⋯ ,  n) is an LI-ideal of the lattice implication algebra L _i  (i = 1,  2, ⋯ ,  n) respectively, then 0 _i  ∈ A _i  (i = 1,  2, ⋯ ,  n) and ( 0 1 , 0 2 , ⋯ , 0 n ) ∈ ∏ i = 1 n A i . For any x , y ∈ ∏ i = 1 n A i , x = ( x 1 , x 2 , ⋯ , x n ) ∈ ∏ i = 1 n A i , y = ( y 1 , y 2 , ⋯ , y n ) ∈ ∏ i = 1 n A i , ∀ z = ( z 1 , z 2 , ⋯ , z n ) ∈ ∏ i = 1 n L i , if (z → x) ′ ⩽ y, we will prove that z = ( z 1 , z 2 , ⋯ , z n ) ∈ ∏ i = 1 n A i . For (z → x) ′ ⩽ y, then (z → x) ′ = ((z₁,  z₂, ⋯ ,  z _n ) → (x₁,  x₂, ⋯ ,  x _n )) ′ = (z₁ → x₁,  z₂ → x₂, ⋯ ,  z _n  → x _n ) ′ = ((z₁ → x₁) ′,   (z₂ → x₂) ′, ⋯ ,   (z _n  → x _n ) ′) ⩽ (y₁,  y₂, ⋯ ,  y _n ) = y, so (z _i  → x _i ) ′ ⩽ y _i , because A _i  (i = 1,  2, ⋯ ,  n) is an LI-ideal of the lattice implication algebra L _i  (i = 1,  2, ⋯ ,  n) respectively, by the Theorem 2.3, z _i  ∈ A _i  (i = 1,  2, ⋯ ,  n), so z = ( z 1 , z 2 , ⋯ , z n ) ∈ ∏ i = 1 n A i , hence ∏ i = 1 n A i is an LI-ideal of the lattice implication product algebra ∏ i = 1 n L i . □

Theorem 4.5.LetL _i  (i = 1,  2, ⋯ ,  n) be lattice implication algebras, ∏ i = 1 n L i their lattice implication product algebra, 0 _i  (i = 1,  2, ⋯ ,  n) is the smallest element ofL _i  (i = 1,  2, ⋯ ,  n). For anyA _i  ⊆ L _i ,   A _i  ≠ {0 _i } (i = 1,  2, ⋯ ,  n), ∏ i = 1 n A i is a prime LI-ideal of the lattice implication product algebra ∏ i = 1 n L i if and only ifA _i  (i = 1,  2, ⋯ ,  n) is a prime LI-ideal of the lattice implication algebraL _i  (i = 1,  2, ⋯ ,  n) respectively.

Proof. “⇒” Because ∏ i = 1 n A i is a prime LI-ideal of the lattice implication product algebra ∏ i = 1 n L i , then ∏ i = 1 n A i is an LI-ideal of the lattice implication product algebra ∏ i = 1 n L i . For any x _i ,  y _i  ∈ L _i  (i = 1,  2, ⋯ ,  n), x = ( x 1 , x 2 , ⋯ , x n ) ∈ ∏ i = 1 n L i , y = ( y 1 , y 2 , ⋯ , y n ) ∈ ∏ i = 1 n L i suppose x _i  ∧ y _i ∈A _i  (i = 1,  2, ⋯ ,  n), then x ∧ y = (x₁ ∧ y₁, x 2 ∧ y 2 , ⋯ , x n ∧ y n ) ∈ ∏ i = 1 n A i . Because ∏ i = 1 n A i is aprime LI-ideal of the lattice implication product algebra ∏ i = 1 n L i , hence x ∈ ∏ i = 1 n A i or y ∈ ∏ i = 1 n A i , x _i  ∈ A _i or y _i  ∈ A _i (i = 1,  2, ⋯ ,  n), so A _i  (i = 1,  2, ⋯ ,  n) is a prime LI-ideal of the lattice implication algebra L _i  (i = 1,  2, ⋯ ,  n) respectively.

“⇐” Because A _i  (i = 1,  2, ⋯ ,  n) is a prime LI-ideal of the lattice implication algebra L _i  (i = 1,  2, ⋯ ,  n) respectively, then A _i is an LI-ideal of the lattice implication algebra L _i  (i = 1,  2, ⋯ ,  n) respectively, By Theorem 4.4, ∏ i = 1 n A i is an LI-ideal of the lattice implication algebra ∏ i = 1 n L i ; For any x = (x₁,  x₂, ⋯ ,  x _n ), y = ( y 1 , y 2 , ⋯ , y n ) ∈ ∏ i = 1 n L i , suppose x ∧ y ∈ ∏ i = 1 n A i , then x ∧ y = ( x 1 ∧ y 1 , x 2 ∧ y 2 , ⋯ , x n ∧ y n ) ∈ ∏ i = 1 n A i , x _i  ∧ y _i  ∈ A _i  (i = 1,  2, ⋯ ,  n). Because A _i  (i = 1,  2, ⋯ ,  n) is a prime LI-ideal of the lattice implication algebra L _i  (i = 1,  2, ⋯ ,  n) respectively, then x _i  ∈ A _i or y _i  ∈ A _i (i = 1,  2, ⋯ ,  n), so x = ( x 1 , x 2 , ⋯ , x n ) ∈ ∏ i = 1 n A i or y = (y₁, y 2 , ⋯ , y n ) ∈ ∏ i = 1 n A i , so ∏ i = 1 n A i is a prime LI-ideal of the lattice implication product algebra ∏ i = 1 n L i . □

Theorem 4.6.LetL _i  (i = 1,  2, ⋯ ,  n) be lattice implication algebras, ∏ i = 1 n L i their lattice implication product algebra. For any A _i  ∈ ∑ (L _i ) and B _i  ⊆ L _i  (i = 1,  2, ⋯ ,  n), then ( ∏ i = 1 n B i ) ( ∏ i = 1 n A i ) = ( ∏ i = 1 n B i ( A i ) ) .

Proof. “⇒” If x ∈ ( ∏ i = 1 n B i ) ( ∏ i = 1 n A i ) , then ∀ b ∈ ∏ i = 1 n B i , x ∧ b ∈ ∏ i = 1 n A i , x = (x₁,  x₂, ⋯,  x _n ), b = (b₁,  b₂, ⋯ ,  b _n ), x ∧ b = (x₁,  x₂, ⋯ ,  x _n ) ∧ (b₁,  b₂, ⋯ ,  b _n ) = (x₁ ∧ b₁,  x₂ ∧ b 2 , ⋯ , x n ∧ b n ) ∈ ∏ i = 1 n A i , so ∀i, x _i  ∧ b _i  ∈ A _i ,  i = 1,  2, ⋯ ,  n, hence x _i  ∈ B _i  (A _i ), x = ( x 1 , x 2 , ⋯ , x n ) ∈ ∏ i = 1 n B i ( A i ) .

“⇐” If x = ( x 1 , x 2 , ⋯ , x n ) ∈ ∏ i = 1 n B i ( A i ) , then x _i  ∈ B _i  (A _i ) (i = 1,  2, ⋯ ,  n), ∀ b = ( b 1 , b 2 , ⋯ , b n ) ∈ ∏ i = 1 n B i , x _i  ∧ b _i  ∈ A _i  (i = 1,  2, ⋯ ,  n), (x₁∧ b₁,  x₂ ∧ b₂, ⋯ ,  x _n  ∧ b _n ) ∈ ∏ i = 1 n A i , x ∧ b ∈ ∏ i = 1 n A i , so x ∈ ( ∏ i = 1 n B i ) ( ∏ i = 1 n A i ) , hence ( ∏ i = 1 n B i ) ( ∏ i = 1 n A i ) = ( ∏ i = 1 n B i ( A i ) ) . □

Theorem 4.7.LetL _i  (i = 1,  2, ⋯ ,  n) be lattice implication algebras, ∏ i = 1 n L i their lattice implication product algebra. For anyA _i is a prime LI-ideal o fL _i , B _i  ⊆ L _i andB _i  (A _i ) ≠ L _i  (i = 1,  2, ⋯ ,  n), then ( ∏ i = 1 n B i ) ( ∏ i = 1 n A i ) = ( ∏ i = 1 n B i ( A i ) ) = ∏ i = 1 n A i .

Proof. It can be obtained by the Theorem 3.6 (3), the Theorem 4.5 and the Theorem 4.6. □

5 Structure of the paper

Lattice implication algebra is an important logical algebra. Based on lattice implication algebra, Xu have proposed lattice-valued logic and their application in uncertainty reasoning and automated reasoning. For the development of lattice-valued logic, we need make clear the structure of lattice implication algebra. As we know, LI-ideal with special properties plays a very important role to study the lattice implication algebra and corresponding lattice-valued logic. A-subset is an important LI-ideal. Therefore, in this paper, we have focused on an A-subset of lattice implication algebras. Firstly, we discussed the properties of the A-subset in lattice implication algebras. Next, we proved that B (A) was an LI-ideal when A was an LI-ideal and A ⊆ B (A) in lattice implication algebras. In the end, we investigated the properties in lattice implication product algebras and proved that the A-subset of ∏ i = 1 n B i was the product of A-subset of B _i  (i = 1,  2, ⋯ ,  n) respectively ( ( ∏ i = 1 n B i ) ( ∏ i = 1 n A i ) = ( ∏ i = 1 n B i ( A i ) ) ) . This work not only enriched the content of LI-ideal of lattice implication algebra. It provides technical and method reference for the study of lattice implication algebra and lattice-valued logic system structure based on LI-ideal content. But also the above work serve as a foundation for further studying lattice-valued logic system based on linguistic truth values lattice implication algebras and developing corresponding applied methods of uncertainty reasoning and resolution automated reasoning with linguistic terms.