Journal of Harbin Institute of Technology(New Series)

Triangle Bounded L-algebras and Triangle Ideals

doi: 10.11916/j.issn.1005-9113.202405

1. School of Science, Xi'an Polytechnic University, Xi'an 710048 , China

2. Business School, Xi’an International University, Xi'an 710077 , China

Funds: Sponsored by Foreign Expert Program of China (Grant No.DL2023041002L) and Yulin City Industry University Research Project (Grant No.CXY-2022-59).

Detailed information

Corresponding author

Xiaolong Xin,Ph.D,Professor.E-mail: xlxin@nwu.edu.cn.

CLC number: O29

Document code: A

Article ID: 1005-9113(2025)05-0052-13

Abstract

This study mainly focuses on the triangle bounded L-algebras and triangle ideals. Firstly, the definition of triangle bounded L-algebras is presented, and several examples with different conditions are outlined along with an exploration of their properties. Moreover, we investigate the structure of triangle bounded L-algebra with a special condition. Secondly, we define the concept of triangle ideals of triangle bounded L-algebra and explore the connection between the triangle ideals of triangle bounded L-algebra L and the ideals of bounded L-algebra E(L). In addition, we classified and studied various classes of triangle ideals, including Stonean triangle ideals, extended Stonean triangle ideals, and lattice ideals, and by introducing the notion of Stonean triangle bounded L algebras, we examine the relationship between Stonean triangle bounded L-algebras and Stonean triangle ideals. Finally, we investigate the interrelationships among these various types of triangle ideals.

Keywords

triangle bounded L-algebra / Stonean triangle bounded L-algebra / triangle ideal / (extended) Stonean triangle ideal / lattice ideal

0 Introduction 1 Preliminaries 2 Triangle Bounded L-algebras 3 Triangle Ideals of Triangle Bounded L-algebras 4 Conclusions

0 Introduction

Non-classical logic turns out to be a valuable device for computers to handle uncertain and fuzzy information. L-algebra has the characteristics of multivalued logic, intuitionistic logic, and quantum logic, therefore, it is closely related to algebraic logic and quantum structure, and is a non-classical logical algebra worth studying. In 2008, Rump^[1] introduced and investigated the L-algebras, exploring their characteristics and properties. Additionally, various special definitions of L-algebras were presented, and it was demonstrated that every L-algebra admits a self-similar closure. In 2012, Rump and Yang^[2] provided an equivalent characterization of intervals in lattice ordered groups that can be represented by L-algebra. In 2019, Wu et al.^[3] explored its relationship with L-algebra by constructing effect algebra. In 2021, Ciungu^[4] provided characterizations of a special L-algebra and explored its relationship with other algebraic structures. In 2023, Kologani^[5] investigated the relationship between the quotient structure of CKL-algebras and BCK-algebras. In 2024, Hu et al.^[6] studied bounded algebras from a different perspective by using a new kind of subalgebra (dual ideals) in bounded L-algebras. In the same year, Yun et al.^[7] studied different ideals of CKL-algebras. Consequently, L-algebras have turned into a focal point of logic algebra study over the past decade.

When studying algebraic structures, triangle algebras are a powerful tool. In 2008, Van Gasse et al.^[8] proposed triangle algebras and proved that there is a connection between these algebras and special residuated lattices. In 2010, Van Gasse et al.^[9] explored various distinct filters of triangle algebras. In 2017, Zahiri et al.^[10] characterized the structure of filters in triangle algebras. In 2021, Zahiri et al.^[11] examined several crucial properties of distinct triangle algebras. In 2024, Zahiri et al.^[12] introduced Stonean algebras and studied the association between the substructure of special triangle algebras. It can be observed that both triangle algebras and filters have a great significance in the study of fuzzy logics and logical algebras. Filters possess characteristics of non-classical algebraic logic and play a significant role in studying the completeness of logical systems^[13-15]. From a logical perspective, all types of filters can be naturally regarded as a set of provable formulas. In fact, the notion of ideals in L-algebras is the same as the concept of filters in normal algebras. Therefore, by studying the structure and properties of triangle algebras and filters, we can extend these results to L-algebras, which is the motivation for our research. Under special conditions, triangle algebras are also triangle bounded L-algebras. That is, triangle bounded L-algebras are a generalization of triangle algebras, we can generalize some results of triangle algebras to triangle bounded L-algebras. Moreover, we can characterize the structure of triangle bounded L-algebras by defining triangle ideals and further study the corresponding logical systems through triangle bounded L-algebra and triangle ideals. By defining Stonean algebras, under special conditions, we can observe that the complement space of a triangle bounded L-algebras forms a complementary lattice.

The primary objective of this study is to explore triangle bounded L-algebras and triangle ideals. Firstly, we extend the definition of bounded L-algebra to the notion of triangle bounded L-algebras by adding a constant u

\notin

{0, 1} and two unary operations ν and μ. We present several examples of triangle bounded L-algebras with different conditions and explore their properties. Moreover, we investigate the connection between triangle bounded L-algebras with special conditions and u, ν, and μ, respectively. Secondly, we introduce the concept of triangle ideals in triangle bounded L-algebras to explore the connection between the triangle ideal I of triangle bounded L-algebra L and the ideal I∩E (L) of bounded L-algebra E (L) . In addition, by introducing the definition of Stonean triangle bounded L-algebras, we study their properties and the connection between Stonean triangle bounded L-algebras and Stonean triangle ideals. We classify and study various classes of triangle ideals, including Stonean triangle ideals, extended Stonean triangle ideals, and lattice ideals. Finally, we investigate the interrelationships among these various types of triangle ideals.

1 Preliminaries

Definition 1.1^[1] For any ω, φ, ψ∈L, if the algebra (L, →, 1) satisfies the following conditions, then it is called an L-algebra.

(L_{1}) 1 \to ω = ω, ω \to ω = ω \to 1 = 1;

(L_{2}) (ω \to φ) \to (ω \to ψ) = (φ ⟶ ω) \to (φ ⟶ ψ);

(L_{3}) ω \to φ = φ \to ω = 1 implies ω = φ .

If the operation → is taken as logical implication, then there is a partial order on L defined by φ≤ψiffφ→ψ=1.

Definition 1.2^[2] An L-algebra is bounded if it contains the smallest element 0. In a bounded L-algebra L, if the operation ′:ω→ω′ is bijective, in which ω′=ω→0 and ω∈L, then we call L a negation ′. The inverse mapping is denoted by ω→ω~.

In Ref.^{[ 5 ]}, the L-algebra L has double negation, if ω′=ω~ for any ω∈L.

Definition 1.3^[4] If the L-algebra L satisfies the condition (K) : ω→ (φ→ω) =1, for any ω, φ∈L, then it is called a KL-algebra.

Definition 1.4^[4] If the L-algebra L satisfies the condition (C) : (φ→ (ψ→ω) ) → (ψ→ (φ→ω) ) =1, for any ψ, ω, φ∈L, then it is called a CL-algebra.

In Ref.^{[ 4 ]}, we can find that the CL-algebra is satisfied with condition (K) .

Example1.1^[5] (L={0, φ, ψ, 1}, →, 0, 1) is a bounded CL-algebra and also a KL-algebra, where the operation → defined on L is presented in Table1.

Table1Cayley table for the binary operation → on L

Definition 1.5^[7] In an L-algebra L, if

\begin{matrix} ((φ \to ψ) \to ω) \to ((ψ \to φ) \to ω) = \\ ((φ \to ψ) \to ω) \to ω \end{matrix}

For every φ, ψ, ω∈L, then it is called a semiregular L-algebra.

Proposition 1.1^[1] In an L-algebra L, if φ≤ψ, then ω→φ≤ω→ψ for any φ, ψ, ω∈L.

Example1.2^[6] (L={0, φ, ψ, 1}, →, 0, 1) is a semiregular L-algebra, as shown in Fig.1, in which the operation → defined on L is presented in Table2.

Fig.1Hasse diagram of L

Table2Cayley table for the binary operation → on L

Proposition 1.2^[8] For any φ, ψ, ω∈L, if L is an L-algebra, then the following conclusions are equivalent.

1) φ≤ψ→φ;

2) φ≤ψ

\Rightarrow

ψ→ω≤φ→ω;

3) ( (φ→ψ) →ω) →ω≤ ( (φ→ψ) →ω) → ( (ψ→φ) →ω) .

We can find that a semiregular L-algebra satisfies the condition (K) .

Definition 1.6^[1] For any φ, ψ∈L, if the subset I of L-algebra L is called an ideal, the following conditions are satisfied:

1) 1∈I;

2) φ, φ→ψ∈I

\Rightarrow

ψ∈I;

3) φ∈I

\Rightarrow

(φ→ψ) →ψ∈I;

4) φ∈I

\Rightarrow

ψ→φ, ψ→ (φ→ψ) ∈I.

In Ref.^[5], we can find that {1} and L are two ideals of L-algebra L. Moreover, if L is a KL-algebra, then 4) can be omitted, and if L is a CL-algebra, then 3) and 4) can be omitted.

Definition 1.7^[10] We call A= (A, ∧, ∨, *,

\Rightarrow

, 0, 1) a residuated lattice, if

1) (A, ∧, ∨, 0, 1) is a bounded lattice;

2) (A, *, 1) is a commutative monoid;

3) φ*ψ≤ω iff φ≤ψ

\Rightarrow

ω for any φ, ψ, ω∈A.

Definition 1.8 ^[16] In a residuated lattice A, if φ* (φ

\Rightarrow

ψ) =φ∧ψ, for any φ, ψ∈A, then A is called a divisible residuated lattice.

Proposition 1.3^[16] A divisible residuated lattice is an L-algebra.

Definition 1.9^[10] The structure A= (A, ∧, ∨, *,

\Rightarrow

, ν, μ, 0, u, 1) is called a triangle algebra, if A can forms a residuated lattice, and for all φ, ψ∈A, we have:

(T.1) νφ≤φ, (T.1′) φ≤μφ

(T.2) νφ≤ννφ, (T.2′) μμφ≤μφ

(T.3) ν (φ∧ψ) =νφ∧νψ, (T.3′) μ (φ∧ψ) = μφ∧μψ

(T.4) ν (φ∨ψ) =νφ∨νψ, (T.4′) μ (φ∨ψ) = μφ∨μψ

(T.5) ν1=1, (T.5′) μ0=0

(T.6) νu=0, (T.6′) μu=1

(T.7) νμφ=μφ, (T.7′) μνφ=νφ

(T.8) ν (φ

\Rightarrow

ψ) ≤νφ

\Rightarrow

νψ

(T.9) (νφ

\Leftrightarrow

νψ) * (μφ

\Leftrightarrow

μψ) ≤ (φ

\Leftrightarrow

ψ) , where φ

\Leftrightarrow

ψ= (φ

\Rightarrow

ψ) * (ψ

\Rightarrow

φ)

(T.10) νφ

\Rightarrow

νψ≤ν (νφ

\Rightarrow

νψ)

2 Triangle Bounded L-algebras

By generalizing the notion of bounded L-algebra, that is, by introducing a constant u (distinct from 0 and 1) and two unary operations v and μ, we obtain the definition of triangle bounded L-algebras. In addition, some examples of triangle bounded L-algebras with different conditions are illustrated, and the properties of these algebras are also studied.

Definition 2.1 The structure (L, →, ν, μ, 0, u, 1) is called a triangle bounded L-algebra, if (L, →, 0, 1) is a bounded L-algebra， and for any φ, ω∈L, we have:

(TL1) νφ≤φ, (TL1′) φ≤μφ

(TL2) νφ≤ννφ, (TL2′) μμφ≤μφ

(TL3) ν1=1, (TL3′) μ0=0

(TL4) νu=0, (TL4′) μu=1

(TL5) νμφ=μφ, (TL5′) μνφ=νφ

(TL6) If φ≤ω, then μφ≤μω

(TL7) ν (φ→ω) ≤νφ→νω

(TL8) If νφ=νω, μφ=μω, then φ=ω

(TL9) νφ→νω≤ν (νφ→νω)

From (TL1) and (TL2) , it is obvious that ννφ=νφ in a triangle bounded L-algebra L. Similarly, from (TL1′) and (TL2′) , we have μμφ=μφ. From (TL1) and (TL9) , νφ→νω=ν (νφ→νω) can be gotten. (TL6) and (TL7) implies that ν and μ are increasing operations. According to the conditions (TL1′) , (TL2′) , and (TL6) , it can be inferred that φ≤μμφ=μφ and if φ≤ω, then μφ≤μω, for any φ, ω∈L. Therefore, μ is a closure on L. According to (TL1) , (TL2) , and (TL7) , it can be inferred that ννφ=νφ≤φ and if φ≤ω, then νφ≤νω, for any φ, ω∈L, Therefore, ν is a dual closure on L.

If the triangle bounded L-algebra L satisfies conditions (K) , (C) , or it is a semiregular L-algebra, then it is called a triangle bounded KL-algebra, triangle bounded CL-algebra, or a triangle bounded semiregular L-algebra, respectively.

Example2.1^[10] Let L= ([0, 1], ∧_L, ∨_L, *→, 0, 1) be a divisible residuated lattice, where φ∨_Lω=max{φ, ω}, φ∧_Lω=min{φ, ω}, φ*ω=min{φ, ω}, if φ≤ω, then φ→ω=1; if ω≤φ, then φ→ω=ω, for any φ, ω∈L.

Then for

I n t (L) = \{[φ_{1}, φ_{2}] : (φ_{1}, φ_{2}) \in L \times L, φ_{1} ⩽ φ_{2}\}

We define:

[φ_{1}, φ_{2}] ⊙ [ω_{1}, ω_{2}] = [φ_{1} * ω_{1}, φ_{2} * ω_{2}]

[φ_{1}, φ_{2}] \Rightarrow [ω_{1}, ω_{2}] = [(φ_{1} \to ω_{1}) \land (φ_{2} \to ω_{2}), φ_{2} \to ω_{2}]

[φ_{1}, φ_{2}] \land [ω_{1}, ω_{2}] = [φ_{1} \land_{L} ω_{1}, φ_{2} \land_{L} ω_{2}]

[φ_{1}, φ_{2}] \lor [ω_{1}, ω_{2}] = [φ_{1} \lor_{L} ω_{1}, φ_{2} \lor_{L} ω_{2}]

For any [φ₁, φ₂], [ω₁, ω₂]∈Int (L) . The structure Int (L) = (Int (L) , ∧, ∨, ⊙,

\Rightarrow

, [0, 0], [1, 1]) is also a residuated lattice.

We also define a partial order relation in Int (L) , [φ₁, φ₂]≤[ω₁, ω₂]

\Leftrightarrow

φ₁≤ω₁ and φ₂≤ω₂, for any [φ₁, φ₂], [ω₁, ω₂] ∈ Int (L) . If we define v[φ₁, φ₂]= [φ₁, φ₁], and μ[φ₁, φ₂]=[φ₂, φ₂], u=[0, 1], for any [φ₁, φ₂]∈Int (L) , then

\begin{matrix} I n t (L) = (I n t (L), \land, \lor, ⊙, \Rightarrow, \\ v, μ, [0, 0], u, [1, 1]) \end{matrix}

is a triangle algebra.

Moreover, we can conclude that Int (L) is also a divisible residuated lattice. This is because, for any [φ₁, φ₂], [ω₁, ω₂]∈Int (L) , if there is an order relation between [φ₁, φ₂] and [ω₁, ω₂], we can get that [φ₁, φ₂]≤[ω₁, ω₂] or [φ₁, φ₂]≥[ω₁, ω₂]. If [φ₁, φ₂]≤[ω₁, ω₂], then

\begin{matrix} [φ_{1}, φ_{2}] \land [ω_{1}, ω_{2}] = \\ [φ_{1} \land_{L} ω_{1}, φ_{2} \land_{L} ω_{2}] = \\ [φ_{1}, φ_{2}] = [φ_{1} * 1, φ_{2} * 1] = [φ_{1}, φ_{2}] ⊙ \\ [1, 1] = [φ_{1}, φ_{2}] ⊙ [(φ_{1} \to ω_{1}) \land_{L} \\ (φ_{2} \to ω_{2}), φ_{2} \to ω_{2}] = [φ_{1}, φ_{2}] ⊙ \\ ([φ_{1}, φ_{2}] \Rightarrow [ω_{1}, ω_{2}]) \end{matrix}

We can use the same method to prove that if

[φ_{1}, φ_{2}] ⩾ [ω_{1}, ω_{2}]

, then

[φ_{1}, φ_{2}] \land [ω_{1}, ω_{2}] = [φ_{1}, φ_{2}] ⊙ ([φ_{1}, φ_{2}] \Rightarrow [ω_{1}, ω_{2}]) .

If there is no order relation between [φ₁, φ₂] and [ω₁, ω₂], then we can get that φ₁＜ω₁ and φ₂＞ω₂, or φ₁＞ω₁ and φ₂＜ω₂. If φ₁＜ω₁ and φ₂＞ω₂, then

\begin{matrix} [φ_{1}, φ_{2}] \land [ω_{1}, ω_{2}] = \\ [φ_{1} \land_{L} ω_{1}, φ_{2} \land_{L} ω_{2}] = \\ [φ_{1}, ω_{2}] = [φ_{1} * ω_{2}, φ_{2} * ω_{2}] = \\ [φ_{1}, φ_{2}] [1 \land_{L} ω_{2}, ω_{2}] = \\ [φ_{1}, φ_{2}] ⊙ [(φ_{1} \to ω_{1}) \land_{L} (φ_{2} \to ω_{2}) \\ φ_{2} \to ω_{2}] = [φ_{1}, φ_{2}] ⊙ \\ ([φ_{1}, φ_{2}] \Rightarrow [ω_{1}, ω_{2}]) \end{matrix}

We can use the same method to prove that when φ₁＞ω₁ and φ₂＜ω₂, we have

\begin{matrix} [φ_{1}, φ_{2}] \land [ω_{1}, ω_{2}] = [φ_{1}, φ_{2}] ⊙ \\ ([φ_{1}, φ_{2}] \Rightarrow [ω_{1}, ω_{2}]) \end{matrix}

Therefore, Int (L) is a divisible residuated lattice (as shown in Fig.2) , according to Definition 1.8.

It is easily evident that Int (L) = (Int (L) , ∧, ∨, ⊙,

\Rightarrow

, v, μ, [0, 0], u, [1, 1]) is also a triangle bounded L-algebras, according to Definition 2.1 and Proposition 1.3.

Consequently, triangle bounded L-algebras are generalization of triangle algebras.

Fig.2Hasse diagram of Int (L)

Remark 2.1 If L= (L, →, ν, μ, 0, u, 1) is a triangle bounded L-algebra, then νφ=νω and μφ=μω iff φ=ω for any φ, ω∈L.

Proof For any φ, ω∈L, according to (TL8) , if νφ=νω and μφ=μω, then φ=ω. If φ=ω, according to (TL6) , then μφ≤μω, μω≤μφ. Therefore, according to (L3) , μφ=μω. And according to (TL3) and (TL7) , there is 1=ν1=ν (φ→ω) ≤νφ→νω and 1=ν1=ν (ω→φ) ≤νω→ νφ. Therefore, according to (L3) , νφ=νω. Consequently, if φ=ω, then νφ=νω and μφ=μω. Therefore, νφ=νω and μφ= μω iff φ= ω for any φ, ω∈L.

In a special case, we can get that a triangle bounded L-algebra solely through the operation ν.

Proposition 2.1 Let (L, →, 0, 1) be a bounded KL-algebra with double negation, u′=u for exists u∈L, and ν satisfies (TL1) - (TL5) and (TL7) - (TL9) . If we define μφ= (νφ′) ′ for any φ∈L, then (L, →, ν, μ, 0, u, 1) is a triangle bounded L-algebra.

Proof Let (L, →, 0, 1) be a bounded KL-algebra with double negation. According to (TL1) , we can get that νφ′≤φ′, and so φ=φ″≤ (νφ′) ′=μφ. So (TL1′) holds.

We can obtain that νφ′=ν (νφ′) =ν (νφ′) ″, according to (TL1) , (TL2) and double negation. Therefore μμφ= (ν (νφ′) ″) ′= (ν (νφ′) ) ′= (νφ′) ′=μφ So (TL2′) holds.

μ 0 = {(ν 0^{'})}^{'} = (ν 1)^{'} = 1^{'} = 0

So (TL3′) holds.

Using the condition u′=u, we can conclude that μu= (νu′) ′= (νu) ′=0′=1, by (TL4) . So (TL4′) holds.

Because νμφ′=μφ′ by (TL5) , we can get that ν (νφ) ′=ν (νφ″) ′= (νφ) ′, and so μνφ= (ν (νφ) ′) ′= (νφ) ″=νφ. So (TL5′) holds.

If φ≤ω, then 1=ν1=ν (ω′→φ′) ≤νω′→νφ′, since ω′≤φ′. So we have νω′≤νφ′. We can conclude that μφ= (νφ′) ′≤ (νω′) ′=μω. So (TL6) holds.

Therefore, (L, →, ν, μ, 0, u, 1) is a triangle bounded L-algebra.

Definition 2.2 In a triangle bounded L-algebra L= (L, →, ν, μ, 0, u, 1) , we set E (L) ={φ∈L|νφ= φ}, ν (L) ={νφ|φ∈L}, and μ (L) ={μφ|φ∈L}.

It can be observed that νφ, μφ∈E (L) , since ννφ=φ and νμφ=μφ, for all φ∈L.

Proposition 2.2 If L= (L, →, ν, μ, 0, u, 1) is a triangle bounded L-algebra, then

E (L) = ν (L) = μ (L) = ν (E (L)) = μ (E (L))

Proof If φ∈E (L)

\subseteq

L, then φ=νφ∈ν (L) for all φ∈L. Therefore, E (L)

\subseteq

ν (L) .

If φ∈ν (L)

\subseteq

L, then φ=νω=μνω∈μ (L) , since (TL5′) and νω∈E (L) , for some ω∈L. Therefore, ν (L)

\subseteq

μ (L) .

If φ∈μ (L) , then φ=μω=νμω∈ν (E (L) ) , since (TL5) and μω∈E (L) , for some ω∈L. Therefore, μ (L)

\subseteq

ν (E (L) ) .

If φ∈ν (E (L) ) , then φ=νω=μνω∈μ (E (L) ) for some ω∈E (L) by (TL5′) . Therefore, ν (E (L) )

\subseteq

μ (E (L) ) .

If φ∈μ (E (L) ) , then φ=μω∈μ (E (L) ) for some ω∈E (L) . We can get that φ=μω∈E (L) , since νμω=μω.

Therefore, μ (E (L) )

\subseteq

E (L) .E (L) =ν (E (L) ) =ν (L) = μ (L) =μ (E (L) ) .

Corollary 2.1 If L is a triangle bounded L-algebra, then E (L) ={φ∈L|μφ=φ} and (E (L) , →, 0, 1) constitutes a bounded L-algebra.

Proof Let L be a triangle bounded L-algebra. For any φ, ψ∈E (L)

\subseteq

L, we have μφ=μνφ=νφ=φ, by Definition 2.1 and 2.2. Therefore, {φ∈L|μφ= φ}. According to (TL1) and (TL9) , we can get that νφ→νψ=φ→ψ and so ν (νφ→νψ) =ν (φ→ψ) . Therefore, φ→ψ=ν (φ→ψ) and so φ→ψ∈E (L) . Moreover, 0, 1∈E (L) , since ν0=0 and ν1=1. Consequently, (E (L) , →, 0, 1) constitutes a bounded L-algebra.

Proposition 2.3 If L is a triangle bounded L-algebra, then νφ=sup{ω∈E (L) |ω≤φ} and μφ=inf{ω∈E (L) |φ≤ω}, for any φ∈L.

Proof If L is a triangle bounded L-algebra. Because ννφ=νφ≤φ for any φ∈L, so νφ∈{ω∈E (L) |ω≤φ}, we have νφ≤sup{ω∈E (L) |ω≤φ}. Moreover, νa≤νφ, since ν is an increasing operation, for every a∈{ω∈E (L) |ω≤φ}. Hence, sup{ω∈E (L) |ω≤φ}≤νφ. Consequently, νφ=sup{ω∈E（L）|ω≤φ}.

We can get that μμφ=μφ≥φ for any φ∈L. Therefore, we have μφ∈{ω∈E（L）|φ≤ω} and μφ≥inf{ω∈E（L）|φ≤ω}. Moreover, inf{ω∈E (L) |φ≤ω}≥μφ, since φ≤a

\Rightarrow

μφ≤μa=a for every a∈{ω∈E (L) |φ≤ω}, by Corollary 2.1. Therefore, μφ=inf{ω∈E (L) |φ≤ω}.

Given a bounded L-algebra, it is possible to generate some triangle bounded L-algebras. Some examples will be given to illustrate it.

Example2.2 If L= ({0, a, 1}, →L, 0, 1) is a bounded L-algebra, then A= (A, →, ν, μ, [0, 0], [0, 1], [1, 1]) is a triangle bounded L-algebra, in which, A={[0, 0], [0, 1], [a, a], [1, 1]}, ν[

ϱ_{1}

ϱ_{2}] = [ϱ_{1}, ϱ_{1}]

, and

μ [ϱ_{1}, ϱ_{2}] = [ϱ_{2}, ϱ_{2}]

, [φ₁, φ₂]∈A. The implication operations defined on L and A are presented in Tables 3 and 4, respectively.

\begin{matrix} [ϱ_{1}, ϱ_{2}] \to [φ_{1}, φ_{2}] = \\ \{\begin{matrix} [ϱ_{1} \to_{L} φ_{1}, ϱ_{2} \to_{L} φ_{2}], \\ i f (ϱ_{1} \to_{L} φ_{1}) ⩽ (ϱ_{2} \to_{L} φ_{2}) \\ [ϱ_{2} \to_{L} φ_{2}, ϱ_{2} \to_{L} φ_{2}], e l s e \end{matrix} \end{matrix}

Table3Cayley table for the implication operation of L

Table4Cayley table for the implication operation of A

It is obvious that

A = (A, \to, [0, 0], [1, 1])

is a bounded L-algebra. Moreover, for any

[ϱ_{1}, ϱ_{2}], [φ_{1}, φ_{2}] \in A

, we can find that

ϱ_{1} ⩽ ϱ_{2} in [ϱ_{1}, ϱ_{2}]

and

[ϱ_{1}, ϱ_{2}]

⩽ [φ_{1}, φ_{2}]

iff

ϱ_{1} ⩽ φ_{1}

and

ϱ_{2} ⩽ φ_{2}

We can conclude that

ν [ϱ_{1}, ϱ_{2}] = [ϱ_{1}, ϱ_{1}] ⩽ [ϱ_{1}, ϱ_{2}], μ [ϱ_{1}, ϱ_{2}] = [ϱ_{2}, ϱ_{2}] ⩾ [ϱ_{1}, ϱ_{2}], ν [ϱ_{1}, ϱ_{2}] = [ϱ_{1},

ϱ_{2}] = ν ν [ϱ_{1}, ϱ_{2}], μ [ϱ_{1}, ϱ_{2}] = [ϱ_{1}, ϱ_{2}] = μ μ [ϱ_{1}, ϱ_{2}], ν [1, 1] = [1, 1], μ [0, 0] = [0, 0], ν μ [ϱ_{1}, ϱ_{2}] = ν [ϱ_{2}, ϱ_{2}] = [ϱ_{2}, ϱ_{2}] = μ [ϱ_{1}, ϱ_{2}], ν [0, 1] = [0, 0], μ [0, 1] = [1, 1], and μ ν [ϱ_{1}, ϱ_{2}] = μ [ϱ_{1}, ϱ_{1}] = [ϱ_{1}, ϱ_{1}] = ν [ϱ_{1}, ϱ_{2}] .

So (TL1) - (TL5) and (TL1′) - (TL5′) holds.

We can observe that in A, if

[ϱ_{1}, ϱ_{2}] ⩽ [φ_{1}, φ_{2}],

then

μ [ϱ_{1}, ϱ_{2}] = [ϱ_{2}, ϱ_{2}] ⩽ [φ_{1}, φ_{2}] = μ [φ_{1}, φ_{2}]

. So (TL6) holds.

For (TL7) . If

ϱ_{1} \to φ_{1} ⩽ ϱ_{2} \to φ_{2}

, then

\begin{matrix} ν ([ϱ_{1}, ϱ_{2}] \to [φ_{1}, φ_{2}]) = ν ([ϱ_{1}, ϱ_{2}] \to [φ_{1}, φ_{2}]) = \\ ν [ϱ_{1} \to_{L} φ_{1}, ϱ_{2} \to_{L} φ_{2}] = [ϱ_{1} \to_{L} φ_{1}, ϱ_{1} \to_{L} φ_{1}] = \\ [ϱ_{1}, ϱ_{1}] \to [φ_{1}, φ_{1}] = ν [ϱ_{1}, ϱ_{1}] \to ν [φ_{1}, φ_{2}] \end{matrix}

ϱ_{1} \to φ_{1} ϱ_{2} \to φ_{2}

, in fact, this situation will result in 0→0

1→0, and 0→a

1→a, because 0→0=1

0=1→0 and 0→a=1

a=1→a.

We can find that

\begin{matrix} ν ([0, 1] \to [0, 0]) = ν [0, 0] = [0, 0] ⩽ [1, 1] = \\ [0, 0] \to [0, 0] = ν [0, 1] \to ν [0, 0] \end{matrix}

and

\begin{matrix} ν ([0, 1] \to [a, a]) = ν [a, a] = [a, a] ⩽ [1, 1] = \\ [0, 0] \to [a, a] = ν [0, 1] \to ν [a, a] \end{matrix}

Therefore,

\begin{matrix} ν ([ϱ_{1}, ϱ_{2}] \to ν [φ_{1}, φ_{2}]) ⩽ \\ ν [ϱ_{1}, ϱ_{2}] \to ν [φ_{1}, φ_{2}] \end{matrix}

for any

[ϱ_{1}, ϱ_{2}], [φ_{1}, φ_{2}] \in A .

So (TL7) holds.

ν [ϱ_{1}, ϱ_{2}] = ν [φ_{1}, φ_{2}], μ [ϱ_{1}, ϱ_{2}] = μ [φ_{1}, φ_{2}],

then

ν [ϱ_{1}, ϱ_{2}] = [ϱ_{1}, ϱ_{1}] = [φ_{1}, φ_{1}] = ν [φ_{1}, φ_{2}]

and

μ [ϱ_{1}, ϱ_{2}] = [ϱ_{2}, ϱ_{2}] = [φ_{2}, φ_{2}] = μ [φ_{1}, φ_{2}]

.Therefore,

[ϱ_{1}, ϱ_{2}] = [φ_{1}, φ_{2}] .

So (TL8) holds.

\begin{matrix} ν [ϱ_{1}, ϱ_{2}] \to ν [φ_{1}, φ_{2}] = [ϱ_{1}, ϱ_{1}] \to [φ_{1}, φ_{1}] = \\ [ϱ_{1} \to φ_{1}, ϱ_{1} \to φ_{1}] = ν [ϱ_{1} \to φ_{1}, Q_{1} \to φ_{1}] = \\ ν ([ϱ_{1}, Q_{1}] \to [φ_{1}, φ_{1}]) = ν (ν [ϱ_{1}, Q_{2}] \to \\ ν [φ_{1}, φ_{2}]) \end{matrix}

So (TL9) holds.

Therefore, we can conclude that A is a triangle bounded L-algebra.

Example2.3 If L= ({0, φ, ψ, ω, 1}, →L, 0, 1) is a bounded L-algebra, then B= (B, →, ν, μ, [0, 0], [0, 1], [1, 1]) is a triangle bounded L-algebra, in which, B={[0, 0], [0, 1], [φ, φ], [ψ, ψ], [ω, ω], [1, 1]}, ν[ψ₁, ψ₂]=[ψ₁, ψ₁], and μ[ψ₁, ψ₂]= [ψ₂, ψ₂], for any [ψ₁, ψ₂], [ω₁, ω₂]∈B. The implication operations defined on L and B are presented in Tables 5 and 6, respectively.

\begin{matrix} [ψ_{1}, ψ_{2}] \to [ω_{1}, ω_{2}] = \\ \{\begin{matrix} [ψ_{1} \to_{L} ω_{1}, ψ_{2} \to_{L} ω_{2}] \\ i f (ψ_{1} \to_{L} ω_{1}) ⩽ (ψ_{2} \to_{L} ω_{2}) \\ [ψ_{2} \to_{L} ω_{2}, ψ_{2} \to_{L} ω_{2}], e l s e \end{matrix} \end{matrix}

Table5Cayley table for the implication operation on L

Table6Cayley table for the implication operation of B

It is obvious that B= (B, →, [0, 0], [1, 1]) is a bounded L-algebra. We can prove that (TL1) - (TL6) , (TL1′) - (TL5′) , and (TL8) - (TL9) hold, according to Example2.2.

For (TL7) . If ψ₁→ω₁≤ψ₂→ω₂, then

\begin{matrix} ν ([ψ_{1}, ψ_{2}] \to [ω_{1}, ω_{2}]) = ν [ψ_{1} \to_{L} ω_{1}, ψ_{2} \to_{L} ω_{2}] = \\ [ψ_{1} \to_{L} ω_{1}, ψ_{1} \to_{L} ω_{1}] = [ψ_{1}, ψ_{1}] \to [ω_{1}, ω_{1}] = \\ ν [ψ_{1}, ψ_{2}] \to ν [ω_{1}, ω_{2}] \end{matrix}

For any [ψ₁, ψ₂], [ω₁, ω₂]∈B. If ψ₁→ω₁

ψ₂→ω₂, in fact, this situation will result in 0→0

1→0 and 0→σ

1→σ, where σ∈{φ, ψ, ω}, because 0→0=1

0=1→0 and 0→σ=1

σ=1→σ. We can get that

\begin{matrix} ν ([0, 1] \to [0, 0]) = ν [0, 0] = [0, 0] ⩽ [1, 1] = \\ [0, 0] \to [0, 0] = ν [0, 1] \to ν [0, 0] \end{matrix}

and

\begin{matrix} ν ([0, 1] \to [σ, σ]) = ν [σ, σ] = [σ, σ] ⩽ [1, 1] = \\ [0, 0] \to [σ, σ] = ν [0, 1] \to ν [σ, σ] \end{matrix}

Therefore,

ν ([ψ_{1}, ψ_{2}] \to [ω_{1}, ω_{2}]) ⩽ ν [ψ_{1}, ψ_{2}] \to ν [ω_{1}, ω_{2}]

For any [ψ₁, ψ₂], [ω₁, ω₂]∈B.So (TL7) holds.

Consequently, B is a triangle bounded L-algebra.

We can give an infinite example of triangle bounded L-algebras, according to Example2.3. If L= ({0, s, t, ..., n, ..., 1}, →_L, 0, 1) is a bounded L-algebra, then B= (B, →, ν, μ, [0, 0], [0, 1], [1, 1]) is a triangle bounded L-algebra, in which,

\begin{matrix} \begin{matrix} B = {[0, 0], [0, 1], [s, s], [t, t], \dots, \\ [n, n], \dots, [1, 1]} \\ ν [φ_{1}, φ_{2}] = [φ_{1}, φ_{1}], and μ [φ_{1}, φ_{2}] = [φ_{2}, \end{matrix} \\ φ_{2}], f o r a n y [φ_{1}, φ_{2}] \in B . \end{matrix}

Example2.4 If L= ({0, a, 1}, →L, 0, 1) is a bounded L-algebra, then C= (C, →, ν, μ, [0, a], [0, 1], [1, 1]) is a triangle bounded L-algebra, where C={[0, a], [a, a], [0, 1], [1, 1]}, ν[0, 1]=ν [0, a]=μ[0, a]=ν[a, a]=[0, a], μ[a, a]=[a, a], and μ[1, 1]=μ[0, 1]=ν[1, 1]=[1, 1]. The implication operations defined on L and C are presented in Tables 7 and 8, respectively.

\begin{matrix} [ψ_{1}, ψ_{2}] \to [ω_{1}, ω_{2}] = \\ \{\begin{matrix} [ψ_{1} \to_{L} ω_{1}, ψ_{2} \to_{L} ω_{2}] \\ i f (ψ_{1} \to_{L} ω_{1}) ⩽ (ψ_{2} \to_{L} ω_{2}) \\ [ψ_{2} \to_{L} ω_{2}, ψ_{2} \to_{L} ω_{2}], e l s e \end{matrix} \end{matrix}

Table7Cayley table for the implication operation of L

Table8Cayley table for the implication operation of C

It is obvious that C= (C, →, [0, a], [1, 1]) is a semiregular L-algebra with negation. Moreover, we can find that ψ₁≤ψ₂ for any [ψ₁, ψ₂]∈C.

We use the same method as in Example2.2, We can find that (TL1) - (TL6) , (TL8) , and (TL1′) - (TL5′) hold.

For any [ψ₁, ψ₂], [ω₁, ω₂]∈C. If ψ₁→ω₁≤ ψ₂→ω₂, then ν ([ψ₁, ψ₂]→[ω₁, ω₂]) =ν[ψ₁→_Lω₁, ψ₂→ _Lω₂]=[ψ₁→_Lω₁, ψ₁→_Lω₁]=[ψ₁, ψ₁]→[ω₁, ω₁]=ν[ψ₁, ψ₂]→ν[ω₁, ω₂]. If ψ₁→ω₁

ψ₂→ω₂, in fact, this situation will result in 0→0

1→a and 0→a

1→a, because 0→0=1

a=1→a and 0→a=1

a=1→a. We can find that

\begin{matrix} ν ([0, 1] \to [0, a]) = ν [a, a] = [0, a] ⩽ [1, 1] = \\ [0, a] \to [0, a] = ν [0, 1] \to ν [0, a] \end{matrix}

and

\begin{matrix} ν ([0, 1] \to [a, a]) = ν [a, a] = [0, a] ⩽ [1, 1] = \\ [0, a] \to [0, a] = ν [0, 1] \to ν [a, a] \end{matrix}

Consequently, ν ([ψ₁, ψ₂]→[ω₁, ω₂]) ≤ν[ψ₁, ψ₂]→ν[ω₁, ω₂]. So (TL7) holds.

For (TL9) , we can obtain that (TL9) holds through verification. Therefore, we can conclude that C= (C, →, ν, μ, [0, a], [0, 1], [1, 1]) is a triangle bounded semiregular L-algebra.

Example2.5 If L= ({0, a, 1}, →_L, 0, 1) is a bounded L-algebra, then D= (D, →, ν, μ, [0, 0], [a, a], [1, 1]) is a triangle bounded L-algebra, in which, D={[0, 0], [a, a], [1, 1]}, ν[0, 0]=μ[0, 0]=ν[a, a]=[0, 0], and ν[1, 1]=μ[1, 1]=μ[a, a]=[1, 1]. The implication operations defined on L and D are presented in Tables 9 and 10, respectively.

Table9Cayley table for the implication operation of L

Table10Cayley table for the implication operation of D

It is obvious that (D, →, [0, 0], [1, 1]) is a bounded L-algebra. We use the same method as in Example2.2, and it can be concluded that D is a triangle bounded L-algebra and it is also a triangle semiregular L-algebra with negation.

In Example2.5, we can find that u′=[a, a]′=[a, a]→[0, 0]=[a, a]=u. Moreover, since (ν[0, 0]′) ′= (ν[1, 1]) ′=[1, 1]′=[0, 0]=μ[0, 0], (ν[1, 1]′) ′= (ν[0, 0]) ′=[0, 0]′=[1, 1]=μ[1, 1], and (ν[a, a]′) ′= (ν[a, a]) ′=[a, a]′=[a, a]= μ[a, a], we can observe that μφ= (νφ′) ′ for any φ∈D. Therefore, this example can be used to explain the existence of the triangle bounded L-algebra given in Proposition 2.1.

In Ref.^[8], a semiregular L-algebra L with negation can form a lattice with meet and join operations, where φ∨ω= (φ~→ω~) →φ, φ∧ω= ( (φ→ω) →φ′) ~for all φ, ω∈L.

We consider that the triangle semiregular L-algebra (A

\subseteq

L×L, →, ν, μ, 0, u, 1) with negation is generated by a semiregular L-algebra L with negation. For any [ψ₁, ψ₂], [ω₁, ω₂]∈A

\subseteq

L×L, in which ψ₁≤ψ₂, ω₁≤ω₂, and we define [ψ₁, ψ₂]≤[ω₁, ω₂]

\Leftrightarrow

ψ₁≤ω₁ and ψ₂≤ω₂. And the operations ν and μ are preserving meet and join operations for components, that is v[ψ₁, ψ₂]∨v[ω₁, ω₂]=v[ψ₁∨ω₁, ψ₂∨ω₂], v[ψ₁, ψ₂]∧v[ω₁, ω₂]=v[ψ₁∧ω₁, ψ₂∧ω₂], μ[ψ₁, ψ₂]∧μ[ω₁, ω₂]=μ[ψ₁∧ω₁, ψ₂∧ω₂], and μ[ψ₁, ψ₂]∨μ[ω₁, ω₂]=μ[ψ₁∨ω₁, ψ₂∨ω₂].

Theorem 2.1 If a semiregular L-algebra L with negation can generate a triangle semiregular L-algebra (A, →, ν, μ, [0, 0], [0, 1], [1, 1]) with negation, then ν[ψ₁, ψ₂]=[ψ₁, ψ₁] and μ[ψ₁, ψ₂]=[ψ₂, ψ₂] for any [ψ₁, ψ₂]∈A.

Proof We first prove that v[ψ₁, ψ₂]=v[ψ₁, 1] for every [ψ₁, ψ₂]∈A. We can conclude that

\begin{matrix} v [ψ_{1}, ψ_{2}] = v [ψ_{1}, ψ_{2}] \lor [0, 0] = \\ v [ψ_{1}, ψ_{2}] \lor v [0, 1] = v [ψ_{1}, 1] \end{matrix}

Suppose ν[ψ₁, ψ₂]=[ω₁, ω₂] and [ω₁, ω₂]≠[ψ₁, ψ₁] for [ω₁, ω₂]∈A. Then [ω₁, ω₂]=ν[ψ₁, ψ₂]=ν[ψ₁, 1]=ν[ψ₁, ψ₁]≤[ψ₁, ψ₁]. We have ω₁＜ψ₁, since ω₁=ψ₁, then ψ₁=ω₁≤ω₂≤ψ₁, this implies that [ω₁, ω₂]=[ψ₁, ψ₁], which is a contradiction.

We can conclude that ν[ψ₁, 1]=ν[ψ₁, ψ₂]=νν[ψ₁, ψ₂]=ν[ω₁, ω₂]=ν[ω₁, 1], and [1, 1]=μ[0, 1]≤μ[ω₁, 1]≤μ[ψ₁, 1], since [0, 1]≤[ω₁, 1]＜[ψ₁, 1]. Therefore, ν[ψ₁, 1]=ν[ω₁, 1] and μ[ψ₁, 1]=μ[ω₁, 1], we have [ψ₁, 1]=[ω₁, 1]. Then ψ₁=ω₁. Therefore, ν[ψ₁, ψ₂]=[ψ₁, ψ₁] for every [ψ₁, ψ₂] in A.

Now, we prove that μ[ψ₁, ψ₂]=[ψ₂, ψ₂] for every [ψ₁, ψ₂] in A. We can get that

\begin{matrix} μ [ψ_{1}, ψ_{2}] = μ [ψ_{1}, ψ_{2}] \land [1, 1] = \\ μ [ψ_{1}, ψ_{2}] \land μ [0, 1] = μ [0, ψ_{2}] \end{matrix}

for every [ψ₁, ψ₂]∈A. Suppose μ[ψ₁, ψ₂]=[c₁, c₂] and [c₁, c₂]≠[ψ₂, ψ₂], for [c₁, c₂] in A. [c₁, c₂]=μ[ψ₁, ψ₂]=μ[0, ψ₂]=μ[ψ₂, ψ₂]≥[ψ₂, ψ₂], we have c₂＞ψ₂. If c₂=ψ₂, then c₁≤c₂=ψ₂≤c₁, which would imply [c₁, c₂]≠ [ψ₂, ψ₂], a contradiction.

We can conclude that μ[0, ψ₂]=μ[ψ₁, ψ₂]=μμ[ψ₁, ψ₂]=μ[c₁, c₂]=μ[0, c₂], and [0, 0]=ν[0, 1]≥ν[0, c₂]≥ν[0, ψ₂], since [0, 1]≥[0, c₂]≥[0, ψ₂]. Hence, ν[0, ψ₂]=ν[0, c₂] and μ[0, ψ₂]=μ[0, c₂], we have [0, ψ₂]=[0, c₂]. Then ψ₂=c₂. Therefore, μ[ψ₁, ψ₂]=[ψ₂, ψ₂] for every [ψ₁, ψ₂] in A.

Therefore, ν[ψ₁, ψ₂]=[ψ₁, ψ₁] and μ[ψ₁, ψ₂]= [ψ₂, ψ₂] for any [ψ₁, ψ₂]∈A.

It is worth noting that in Theorem 2.1, if [ω₁, ω₂]∈A, then we also require [ω₁, 1], [0, ω₂], [ω₁, ω₁], [ω₂, ω₂]∈A.

In an algebra X with join and meet operations, ψ∈X is called meet irreducible, if for any ψ₁, ψ₂∈X such that ψ₁∧ψ₂=ψ, then ψ₁=ψ or ψ₂=ψ. Similarly, ψ∈X is called join irreducible if for any ψ₁, ψ₂∈X, ψ₁∨ψ₂=ψ, then ψ₁=ψ or ψ₂=ψ.

Proposition 2.4 In a triangle semiregular L-algebra (A, →, ν, μ, [0, 0], u, ^{[1, 1]}) with negation, which is generated by a semiregular L-algebra L with negation. If [0, 0] is meet irreducible and [1, 1] is join irreducible, then u=[0, 1] or u=[ψ, ψ] for some ψ∈L.

Proof Let u=[ψ₁, ψ₂] for some [ψ₁, ψ₂]∈A. If ψ₁=ψ₂, then u [ψ₁, ψ₁]. If ψ₁＜ψ₂, then u≠[ψ₂, ψ₂] and u≠[ψ₁, ψ₁]. We have v[ψ₂, ψ₂]≠vu= [0, 0]or μ[ψ₁, ψ₁]≠μu=[1, 1].

We can conclude ν[ψ₂, ψ₂]＞ν[ψ₁, ψ₂]=νu=[0, 0], since [1, 1]=μu=μ[ψ₁, ψ₂]＜μ[ψ₂, ψ₂], because μ and v are increasing. We can use the same method to prove μ[ψ₁, ψ₁]＜[1, 1]. Therefore, we can conclude that ν[ψ₂, ψ₂]≠[0, 0] and μ[ψ₁, ψ₁]≠ [1, 1]. Since ν[0, ψ₂]≤ν[ψ₁, ψ₂]=[0, 0], it follows that

\begin{matrix} ν [0, 1] \land ν [ψ_{2}, ψ_{2}] = ν ([0, 1] \land \\ [ψ_{2}, ψ_{2}]) = ν [0, ψ_{2}] = [0, 0] \end{matrix}

since [0, 0] is irreducible. Therefore, ν[0, 1]=[0, 0].

Since

\begin{matrix} [1, 1] = μ [ψ_{1}, ψ_{2}] ⩽ μ [ψ_{1}, 1] = μ ([0, 1] \lor \\ [ψ_{1}, ψ_{1}]) = μ [0, 1] \lor μ [ψ_{1}, ψ_{1}] \end{matrix}

and [1, 1] is join irreducible, we can conclude that [1, 1]=μ[0, 1].

This means that v[0, 1]=vu and μ[0, 1]=μu, so we obtain u=[0, 1].

Corollary 2.2 If a linear semiregular L-algebra L with negation can generate a triangle semiregular L-algebra (A, →, ν, μ, [0, 0], u, [1, 1]) with negation, in which A

\subseteq

L×L contains more than four elements, then u=[0, 1].

Proof We can find that (A, →, [0, 0], [1, 1]) is a linear semiregular L-algebra with negation, [0, 0] is meet irreducible and [1, 1] is join irreducible. According to Proposition 2.4, we can find that u=[ψ, ψ] or u=[0, 1], where 0＜ψ＜1 and ψ∈L.

Now, we want to prove that u=[ψ, ψ] does not hold. Since A has more than four elements, for ω∈L{0, ψ, 1}, 0＜ψ＜ω＜1 or 0＜ω＜ψ＜1. When 0＜ψ＜ω＜1, we have [1, 1]=μ[ψ, ψ]≤μ[ψ, ω]. So μ[ψ, ω]=μ[ψ, ψ]. For ν[ψ, ω], we have ν[ψ, ω]

[0, ψ] or ν[ψ, ω]≤[0, ψ]. ν[ψ, ω]≤ [0, ψ] does not hold, since [0, 1]≤μ[0, 1]= μ[ψ, ψ]∧μ[0, 1]=μ[0, ψ]≤μν[ψ, ω]=ν[ψ, ω]≤ [ψ, ω], and so ω＜1. Hence, ν[ψ, ω]

[0, ψ]. Consequently, ν[ψ, ω]=νν[ψ, ω]＜ν[0, ψ]≤ν[ψ, ψ]=[0, 0]. Then ν[ψ, ω]=ν[ψ, ψ]. Since μ[ψ, ω]=μ[ψ, ψ], we can get that [ψ, ω]=[ψ, ψ], a contradiction.

We can use the same method to prove when 0＜ω＜ψ＜1, u=[ψ, ψ] does not hold. Therefore, we can conclude that u=[0, 1].

Example2.6 We set L= (L, →, ν, μ, [0, 0], [0, 1], [1, 1]) , in which, L={[0, 0], [0, a], [0, 1], [1, 1]}, ν[0, 0]=μ[0, 0]=ν[0, 1]=ν[0, a]=[0, 0], μ[0, a]=[0, a], ν[1, 1]=μ[1, 1]=μ[0, 1]=[1, 1].

The operation → of L is shown in Table11.

Table11Cayley table for the implication operation of L

It is obvious that L= (L, →, [0, 0], [1, 1]) is a linear bounded semiregular L-algebra with negation. Moreover, we can find that ψ₁≤ψ₂ for any [ψ₁, ψ₂]∈L.

It is easy to verify (TL1) - (TL6) , (TL8) , and (TL1′) - (TL5′) hold.

For (TL7) . We can find that ν ([0, 1]→[0, a]) = ν[0, a]=[0, 0]≤[1, 1]=ν[0, 1]→ν[0, a], ν ([0, 1]→[0, 0]) =ν[0, a]=[0, 0]≤[1, 1]= ν[0, 1]→ν[0, 0], and ν ([0, a]→[0, 0]) =ν[0, 1]= [0, 0] ≤[1, 1]= ν[0, a]→ν[0, 0]. The same methods are used to prove that ν ([ψ₁, ψ₂]→[ω₁, ω₂]) ≤ν[ψ₁, ψ₂]→ν[ω₁, ω₂] for any [ψ₁, ψ₂], [ω₁, ω₂]∈L. So (TL7) holds.

For (TL9) , we can obtain that (TL9) holds through verification.

Therefore, we can conclude that L= (L, →, ν, μ, [0, a], [0, 1], [1, 1]) is a linear triangle bounded semiregular L-algebra.

3 Triangle Ideals of Triangle Bounded L-algebras

In this section, we propose and explore triangle ideals of different types of triangle bounded L-algebras.

Definition 3.1The subset I of triangle bounded L-algebra L is referred to an triangle ideal if for any φ, ψ∈L, the following conditions are satisfied:

1) 1∈I;

2) If φ, φ→ψ∈I, then ψ∈I;

3) If φ∈I, then (φ→ψ) →ψ∈I;

4) If φ∈I, then ψ→φ, ψ→ (φ→ψ) ∈I;

5) If φ∈I, then νφ∈I.

In a triangle ideal I, we have μφ∈I if φ∈I, since φ≤μφ and Definition 3.1. We can easily discover that the triangle ideals of triangle bounded L-algebra L are special ideals of L-algebra, and {1} and L are two triangle ideals and they are trivial. It can be observed that when L can constitutes a triangle bounded KL-algebra, the set I is a triangle ideal if it only needs to satisfy conditions 1) , 2) , 3) , and 5) . If L can constitutes a triangle bounded CL-algebra, then I is a triangle ideal if it only needs to satisfy conditions 1) , 2) , and 5) .

Example3.1In Example2.2, I={[a, a], [1, 1]} is fulfilled with the conditions of the triangle ideal of a triangle bounded L-algebra. Thus, it is a triangle ideal of the triangle bounded L-algebra A.

Proposition 3.1Let (L, →, ν, μ, 0, u, 1) be a triangle bounded L-algebra. If I is a triangle ideal of the triangle bounded L-algebra L, then φ∈I

\Leftrightarrow

νφ∈I and I∩E (L) are ideals of the bounded L-algebra E (L) .

Proof Let I be a triangle ideal of the triangle bounded L-algebra L. We have φ∈I

\Leftrightarrow

νφ∈I, according to Definition 3.1 and (TL1) . It can be obtained that E (L) is a bounded L-algebra, according to Corollary 2.1. It is obvious that E (L) is a trivial ideal of E (L) . We can examine that I∩E (L) is satisfied with conditions of ideal of L-algebra, for any φ, ω∈E (L)

\subseteq

It is straightforward to examine that 1∈I∩E (L) . If φ, φ→ω∈I∩E (L) , then φ, φ→ω∈I and φ, φ→ω∈E (L) . Since I is a triangle ideal of L and E (L) is an ideal of E (L) , ω∈I and ω∈E (L) . Therefore, ω∈I∩E (L) .

If φ∈I∩E (L) , then φ∈I and φ∈E (L) . We can get that (φ→ω) →ω∈I and ω→φ, ω→ (φ→ω) ∈I, since I is a triangle ideal of L. Moreover, we have (φ→ω) →ω∈E (L) and ω→φ, ω→ (φ→ω) ∈ E (L) , since E (L) is a bounded L-algebra. Therefore, (φ→ω) →ω∈I∩E (L) and ω→φ, ω→ (φ→ω) ∈ I∩E (L) .

Therefore, I∩E (L) is an ideal of E (L) .

Proposition 3.2Let (L, →, ν, μ, 0, u, 1) be a triangle bounded CL-algebra. If φ∈I

\Leftrightarrow

νφ∈I and I∩E (L) are ideals of the bounded L-algebra E (L) , then I

\subseteq

L is a triangle ideal of the triangle bounded CL-algebra L.

Proof If L is a triangle bounded CL-algebra and I∩E (L) is an ideal of E (L) , and φ∈I

\Leftrightarrow

νφ∈I. Therefore, we only need to demonstrate that conditions 1) and 2) in Definition 3.1 are satisfied. We can find that 1∈I∩E (L)

\subseteq

I. For any φ, ω∈L, if φ, φ→ω∈I, then νφ, ν (φ→ω) ∈I. Since νφ, ν (φ→ω) ∈E (L) , νφ, ν (φ→ω) ∈I∩E (L) . Because (νφ→νω) ∈E (L) and ν (φ→ω) → (νφ→νω) =1∈I∩E (L) , we can get that (νφ→νω) ∈I∩E (L) . Because νφ∈I∩E (L) , we have νω∈I∩E (L)

\subseteq

I, and so ω∈I.

Therefore, I

\subseteq

L is triangle ideal of the triangle bounded CL-algebra L.

Theorem 3.1If (L, →, ν, μ, 0, u, 1) is a triangle bounded CL-algebra and I

\subseteq

L, then φ∈I

\Leftrightarrow

νφ∈I and I∩E (L) are ideals of the bounded L-algebra E (L) iff I is a triangle ideal of L.

Proof According to Propositions 3.1 and 3.2, we can get that φ∈I

\Leftrightarrow

νφ∈I and I∩E (L) are ideals of the bounded L-algebra E (L) iff I is a triangle ideal of the triangle bounded CL-algebra L.

Definition 3.2A bounded L-algebra L with negation is regarded as a Stonean bounded L-algebra if the supremum of φ′ and φ″ is 1, for any φ∈L.

Definition 3.3A triangle bounded L-algebra L is regarded as a Stonean triangle bounded L-algebra if L is a Stonean bounded L-algebra.

In Ref.^[2], if L is a KL-algebra with negation and it satisfied condition φ≤ω iff ω′≤φ′ for any φ, ω∈L. Then L can define the meet and join operations.

Example3.2Let

\begin{matrix} L = ({[0, a], [a, a], [0, 1], [a, 1]}, \to, \\ ν, μ, [0, a], [0, 1], [a, 1]) \end{matrix}

with Hasse diagram as shown in Fig.3, in which, 0≤a≤1, ν[0, 1]=ν[0, a]=μ[0, a]=ν[a, a]=[0, a], μ[a, a]=[a, a], and ν[a, 1]=μ[a, 1]=μ[0, 1]=[a, 1]. The operation → defined on L is shown in Table12.

Fig.3Hasse diagram of L

Table12Cayley table for the implication operation of L

It is obvious that

\begin{matrix} L = ({[0, a], [a, a], [0, 1], [a, 1]}, \to, \\ ν, μ, [0, a], [0, 1], [a, 1]) \end{matrix}

It is a triangle bounded KL-algebra with negation and it satisfied condition x≤y iff y′≤x′, for all x, y∈L. Moreover, we find[0, a]′∨[0, a]″=[a, 1]∨[0, a]=[a, 1], [a, a]′∨[a, a]″=[0, 1]∨ [a, a]=[a, 1], [0, 1]′∨[0, 1]″=[a, a]∨[1, 1]= [a, 1], and [a, 1]′∨[a, 1]″= [0, a]∨[a, 1]=[a, 1]. Therefore,

{[ϱ_{1}, ϱ_{2}]}^{'} \lor {[ϱ_{1}, ϱ_{2}]}^{''} = [a, 1]

, for any

[ϱ_{1}, ϱ_{2}]

∈C.

It can be concluded that L is a Stonean triangle bounded L-algebra.

Example3.3Let L= (L, →, ν, μ, [0, 0], [a, a], [1, 1]) be a triangle bounded L-algebra with Hasse diagram as shown in Fig.4, in which, L={[0, 0], [a, a], [1, 1]}, ν[0, 0]=μ[0, 0]=ν[a, a]=[0, 0], ν^{[1, 1]}=μ[1, 1]=μ[a, a]=[1, 1]. The operation → of L is presented in Table13.

Fig.4Hasse diagram of L

Table13Cayley table for the implication operation of L

It is obvious that L is a triangle bounded L-algebra. However, the supremum of [a, a]′=[a, a] and [a, a]″=[a, a] is [a, a], not ^{[1, 1]}.

Therefore, L is not a Stonean triangle bounded L-algebra.

Proposition 3.3If L is a Stonean triangle bounded KL-algebra, then E (L) ={φ∈L|νφ=φ} and S={φ∈L| (νφ) ′=0} are Stonean bounded KL-algebras and are closed with respect to the operations ν and μ on L.

Proof Since L is a Stonean triangle bounded KL-algebra, φ→ω=νφ→νω=ν (νφ→νω) =ν (φ→ ω) , for any φ, ω∈E (L)

\subseteq

L. Therefore, E (L) ={φ∈L|νφ= φ} is closed under the operation → in L. For any φ∈E (L) , we have νφ=φ∈E (L) and μφ=φ∈E (L) , according to Corollary 2.1. Consequently, we can get that E (L) ={φ∈L|νφ=φ} is a Stonean bounded KL-algebra and is closed under the operations ν and μ.

According to Proposition 1.1 and ω≤φ→ω, we can find that (ν (φ→ω) ) ′≤ (νω) ′=0, for every φ, ω∈S

\subseteq

L. Therefore, it can be concluded that φ→ω∈S and S={φ∈ L| (νφ) ′=0} are closed under the operation → in L. Moreover, S={φ∈L| (νφ) ′=0}is a Stonean bounded KL-algebra. For any φ∈S, we have (ννφ) ′= (νφ) ′= 0, since ννφ= νφ and (νφ) ′=0. Therefore, vφ∈S. Moreover, νφ≤νμφ, since φ≤μφ. Therefore, (νμφ) ′≤ (νφ) ′=0. We can conclude that (νμφ) ′=0. Consequently μφ∈S. As a result, S={φ∈L| (νφ) ′=0} is closed under the operations ν and μ on L.

Proposition 3.4In a triangle bounded KL-algebra L with negation, we will demonstrate that the following conclusions hold and are equivalent.

1) (ν (φ→ω) ) ′= (ν (ω→φ) ) ′ for any φ, ω∈L{0}.

2) (νω→νφ) ′= (νφ→νω) ′ for any φ, ω∈L{0}.

3) {φ∈L| (νφ) ′=0}=L{0}.

Proof 1) 3) : For every φ, ω∈L\{0}, we can get that (ν (ω→φ) ) ′= (ν (φ→ω) ) ′. When ω= 1, (ν (1→φ) ) ′= (ν (φ→1) ) ′, that is (νφ) ′= (ν1) ′= 1′=0. It can be concluded that L\{0}

\subseteq

{φ∈L| (νφ) ′= 0}. Moreover, it is obvious that {φ∈ L| (νφ) ′=0}

\subseteq

L\{0}. Therefore, {φ∈L| (νφ) ′= 0}=L\{0}.

\Rightarrow

1) :For any φ, ω∈L\{0}, if {φ∈ L| (νφ) ′=0}=L\{0}, then φ, ω∈{φ∈L| (νφ) ′= 0}. We have (ν (φ→ω) ) ′≤ (νω) ′=0, since ω≤φ→ω and Proposition 1.1. So, (ν (φ→ω) ) ′=0. We can use the same method to obtain (ν (ω→φ) ) ′=0. Therefore, (ν (φ→ω) ) ′= (ν (ω→φ) ) ′ for any φ, ω∈L\{0}.

\Rightarrow

3) : Let (νω→νφ) ′= (νφ→νω) ′ for any φ, ω∈L\{0}. If ω=1, then

\begin{matrix} (ν ω \to ν φ)^{'} = (ν φ \to ν ω)^{'} \Rightarrow \\ (ν 1 \to ν φ)^{'} = (ν φ \to ν 1)^{'} \Rightarrow \\ (ν φ)^{'} = 1^{'} = 0 \end{matrix}

We can conclude that L\{0}

\subseteq

{φ∈L| (νφ) ′=0}. Moreover, it is obvious that {φ∈L| (νφ) ′=0}

\subseteq

L\{0}. Therefore, {φ∈L| (νφ) ′=0}=L\{0}.

\Rightarrow

2) ： Let {φ∈L| (νφ) ′=0}=L\{0}. Then for any φ, ω∈L\{0}={φ∈L| (νφ) ′=0}, we have (νω→νφ) ′≤ (νφ) ′=0, since νφ≤νω→νφ and Proposition 1.1. Therefore, (νω→νφ) ′=0. We can use the same method to obtain (νφ→νω) ′=0. Therefore, (νω→νφ) ′= (νφ→νω) ′ for any φ, ω∈L\{0}.

Theorem 3.2If the triangle bounded KL-algebra L satisfies the condition (νω→νφ) ′= (νφ→ νω) ′, for any φ, ω∈L\{0}, then L is a Stonean triangle bounded L-algebra.

Proof According to Proposition 3.4, if (νφ→νω) ′= (νω→νφ) ′, for any φ, ω∈L\{0}, then {φ∈L| (νφ) ′=0}=L\{0}. We can conclude that φ′≤ (νφ) ′=0, since (TL1) and Proposition 1.1. Hence, φ′=0 and so φ″=1. Therefore, the supremum of φ′ and φ″ is 1, for any φ∈L. Then L is a Stonean triangle bounded L-algebra.

When L is a Stonean triangle bounded L-algebra and L′={φ′|φ∈L} is a complement space of L. Then we can observe that the supremum of φ and φ′ is 1, for any φ∈L′. If the infimum of φ and φ′ is 0, for any φ∈L′, then L′ forms a complementary lattice. In the following text, we will refer to Stonean triangle bounded L-algebra as the Stonean algebra, without causing any ambiguity.

Corollary 3.1If the triangle bounded KL-algebra L satisfies one of the conditions in Proposition 3.4, then L is a Stonean algebra.

Definition 3.4In a triangle bounded L-algebra L with negation, the triangle ideal I of L is a Stonean triangle ideal if the supremum of φ′ and φ″ belongs to I under ν operation, for any φ∈L.

Definition 3.5In a triangle bounded L-algebra L with negation, the triangle ideal I is an extended Stonean triangle ideal if the supremum of (vφ) ′ and (vφ) ″ belongs to I, for any φ∈L.

Remark 3.1Let L be a triangle bounded KL-algebra with negation and join operation. If I is a Stonean triangle ideal of L and ν (φ∨ω) =νφ∨νω for any φ, ω∈L, then I is an extended Stonean triangle ideal.

ProofThe supremum of φ′ and φ″ belongs to I under ν operation, that is ν (φ″∨φ′) ∈I, for all φ∈L, since I is a Stonean triangle ideal. Moreover,

\begin{matrix} ν (φ^{''} ⋁ φ^{'}) = ν (φ^{'}) ⋁ \\ ν (φ^{''}) ⩽ (ν φ)^{''} ⋁ (ν φ)^{'} \end{matrix}

according to (TL7) . Therefore, ν (φ″∨φ′) → ( (νφ) ′∨ (νφ) ″) =1∈I. Since ν (φ″∨φ′) ∈I, we have (νφ) ′∨ (νφ) ″∈I. Hence, we can get that the supremum of (vφ) ′ and (vφ) ″ belongs to I, for any φ∈L. Therefore, I is an extended Stonean triangle ideal.

Example3.4In Example3.2, it is clear that I={[a, 1]} is fulfilled with the conditions of the ideal of a triangle bounded L-algebra. Thus, it is a triangle ideal of

\begin{matrix} L = ({[0, a], [a, a], [0, 1], [a, 1]}, \to, \\ ν, μ, [0, a], [0, 1], [a, 1]) \end{matrix}

Moreover, we can find that {[a, 1]} is a Stonean triangle ideal of L and {[a, 1]} is also an extended Stonean triangle ideal.

Example3.5Let L= ({0, a, 1}, →, 0, a, 1) be a triangle bounded L-algebra, in which, μ0=ν0=νa=0 and ν1=μ1=μa=1. The operation → of L is presented in Table14.

Table14Cayley table for the implication operation of L

We can prove that I={1} is an extended Stonean triangle ideal of L. However, we can find that the supremum of a′=a and a″=a do not belong to I. Consequently, I={1} is not a Stonean triangle ideal of L.

Theorem 3.3Let L be a Stonean triangle bounded L-algebra. If I is a triangle ideal of L, then I is a Stonean triangle ideal of L.

Proof When L is a Stonean triangle bounded L-algebra and I is a triangle ideal of L. For all φ∈L, The supremum of φ′ and φ″ is 1, and so the supremum of φ′ and φ″ is also 1 under ν operation, since ν1=1 and 1∈I. Consequently, I is a Stonean triangle ideal of L.

Proposition 3.5In a triangle bounded L-algebra L with negation. If {1} is a Stonean triangle ideal of L, then L is Stonean algebra.

Proof Let {1} be a triangle ideal of L. Then the supremum of φ′ and φ″ belongs to {1} under ν operation, for any φ∈L. Since νφ≤φ and ν1=1, it can be concluded that the supremum of φ′ and φ″ is 1. Hence, L is a Stonean triangle bounded L-algebra.

Corollary 3.2In a triangle bounded L-algebra L with negation, {1} is a Stonean triangle ideal of L iff L is Stonean algebra.

Theorem 3.4Let I be an extended Stonean triangle ideal of the triangle bounded L-algebra L and J be a triangle ideal of L, where I

\subseteq

J. Then J is an extended Stonean triangle ideal of L.

Proof Let I be an extended Stonean triangle ideal of L. Then the supremum of (νφ) ′ and (νφ) ″ belongs to I, for all φ∈L. So, it can be concluded that the supremum of (νφ) ′ and (νφ) ″ belongs to J, since I

\subseteq

J. Consequently, J is an extended Stonean triangle ideal in L.

Corollary 3.3Let L be a triangle bounded L-algebra with negation. {1} is an extended Stonean triangle ideal of L iff every triangle ideal of L is also an extended Stonean triangle ideal.

Proposition 3.6In a Stonean triangle bounded KL-algebra L with join operation. If ν (φ∨ω) =νφ∨νω for any φ, ω∈L, then triangle ideal I of L is also an extended Stonean triangle ideal.

Proof Let L be a Stonean triangle bounded KL-algebra with join operation, then {1} is a Stonean triangle ideal of L, according to Corollary 3.2. We can conclude that {1} is an extended Stonean triangle ideal of L by Remark 3.1. Therefore, every triangle ideal I is an extended Stonean triangle ideal in L, according to Corollary 3.3.

Proposition 3.7If I is a triangle ideal of a triangle bounded KL-algebra L with double negation and if φ∈L, then φ∈I or φ′∈I, then I is an extended Stonean triangle ideal.

Proof For every φ∈L. When φ∈I, we have φ″∈I, since νφ∈I and νφ= (νφ) ″. Therefore, the supremum of (νφ) ′ and (νφ) ″ belongs to I. When φ

\notin

I, we have φ′∈I, and so (νφ) ′∈L, since φ′≤ (νφ) ′. Thus the supremum of (νφ) ′ and (νφ) ″ belongs to I. Consequently, I is an extended Stonean triangle ideal.

Definition 3.6For any φ, ω∈L, if the subset I of a triangle bounded L-algebra L with the meet operation is referred to as a lattice ideal, the following conditions are satisfied:

1) If φ∈I and φ≤ω, then ω∈I;

2) If φ, ω∈I, then φ∧ω∈I;

3) If φ∈I, then νφ∈I.

Example3.6In Example3.2, it is clear that I={[a, 1]} is fulfilled with the conditions of the lattice ideal. Thus, I={[a, 1]} is also a lattice ideal.

Definition 3.7 ^[2]We refer to (S, *, →, 1) as a left hoop if the following conditions are satisfied for any φ, ψ, ω∈S:

1) (S, *, 1) is a monoid;

2) φ→φ=1;

3) (φ*ψ) →ω=φ→ (φ→ω) ;

4) (ψ→ω) *ψ= (ω→ψ) *ω.

In Ref.^[8], a semiregular L-algebra L with negation is a left hoop. Hence, we have φ*ω≤φ∧ω for any φ, ω∈L, since φ≤ω→φ and φ≤ω→ω=1.

Proposition 3.8If L is a triangle semiregular L-algebra with negation, then a triangle ideal of L is a lattice ideal.

Proof If I is a triangle ideal of a triangle semiregular L-algebra L with negation, then I holds on conditions 1) and 3) of Definition 3.6. We only need to prove that φ∧ω∈I for any φ, ω∈I. Since φ*ω≤ φ∧ω, φ≤ω→φ∧ω. So φ→ (ω→φ∧ω) =1∈ I. Since I is a triangle ideal, ω→φ∧ω∈I. Hence, φ∧ω∈I, since ω∈I. Therefore, I can form a lattice ideal of L.

4 Conclusions

Triangle bounded L-algebras and ideals have a great significance in the study of fuzzy logics and logical algebras. This paper focuses on exploring triangle bounded L-algebras and triangle ideals. Firstly, we extend the definition of bounded L-algebra to the notion of triangle bounded L-algebras by adding a constant u

\notin

{0, 1} and two unary operations ν and μ. Secondly, we defined the notion of triangle ideals of triangle bounded L-algebras to explore the connection between the triangle ideal I of triangle bounded L-algebra L and the ideal I∩E (L) of bounded L-algebra E (L) . In addition, by introducing the concept of Stonean triangle bounded L-algebra, we study its properties and the connection between Stonean triangle algebras and (extended) Stonean triangle ideals. Various classes of triangle ideals, including Stonean triangle ideals, extended Stonean triangle ideals, and lattice ideals, are introduced and studied. Finally, the interrelationships among various types of ideals are investigated. In the future, we will characterize the structure of triangle bounded L-algebras by studying closure operators, interior operators, and local bounded L-algebras, and further investigate closure operators and interior operators in Stonean spaces.

Fig.1Hasse diagram of L

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Fig.2Hasse diagram of Int (L)

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Fig.3Hasse diagram of L

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Fig.4Hasse diagram of L

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Table1Cayley table for the binary operation → on L

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Table2Cayley table for the binary operation → on L

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Table3Cayley table for the implication operation of L

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Table4Cayley table for the implication operation of A

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Table5Cayley table for the implication operation on L

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Table6Cayley table for the implication operation of B

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Table7Cayley table for the implication operation of L

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Table8Cayley table for the implication operation of C

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Table9Cayley table for the implication operation of L

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Table10Cayley table for the implication operation of D

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Table11Cayley table for the implication operation of L

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Table12Cayley table for the implication operation of L

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Table13Cayley table for the implication operation of L

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Table14Cayley table for the implication operation of L

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Fig.1Hasse diagram of L

Fig.2Hasse diagram of Int (L)

Fig.3Hasse diagram of L

Fig.4Hasse diagram of L

Table1Cayley table for the binary operation → on L

Table2Cayley table for the binary operation → on L

Table3Cayley table for the implication operation of L

Table4Cayley table for the implication operation of A

Table5Cayley table for the implication operation on L

Table6Cayley table for the implication operation of B

Table7Cayley table for the implication operation of L

Table8Cayley table for the implication operation of C

Table9Cayley table for the implication operation of L

Table10Cayley table for the implication operation of D

Table11Cayley table for the implication operation of L

Table12Cayley table for the implication operation of L

Table13Cayley table for the implication operation of L

Table14Cayley table for the implication operation of L

Image(4) / Table(14)

Citation

Hao Chen,Xiaolong Xin.Triangle bounded L-algebras and triangle ideals.Journal of Harbin Institute of Technology (New Series),2025,32(5):52-64.

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Fig.1Hasse diagram of L

Fig.2Hasse diagram of Int (L)

Fig.3Hasse diagram of L

Fig.4Hasse diagram of L

Table1Cayley table for the binary operation → on L

Table2Cayley table for the binary operation → on L

Table3Cayley table for the implication operation of L

Table4Cayley table for the implication operation of A

Table5Cayley table for the implication operation on L

Table6Cayley table for the implication operation of B

Table7Cayley table for the implication operation of L

Table8Cayley table for the implication operation of C

Table9Cayley table for the implication operation of L

Table10Cayley table for the implication operation of D

Table11Cayley table for the implication operation of L

Table12Cayley table for the implication operation of L

Table13Cayley table for the implication operation of L

Table14Cayley table for the implication operation of L

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0 Introduction

1 Preliminaries

2 Triangle Bounded L-algebras

3 Triangle Ideals of Triangle Bounded L-algebras

4 Conclusions

LINKS