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2046-1402

F1000 Research Limited

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10.12688/f1000research.159175.1

Research Article

Articles

Li-ideals of implicative almost distributive lattices

[version 1; peer review: 1 approved, 1 approved with reservations]

Mechderso

Alachew Amaneh

Conceptualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0009-0007-9566-9901 a 1 Munie

Tilahun Mekonnen

Conceptualization Visualization Writing – Review & Editing 2 1Department of Mathematics, University of KabriDahar, Kabri Dahar, Somalie, Ethiopia 2Department of Mathematics, Bahir Dar University College of Science, Bahir Dar, Amhara, 6000, Ethiopia

a alachewamaneh@gmail.com

No competing interests were disclosed.

10 2 2025

2025

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24 12 2024

2025

This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Background

In this paper, we introduce the concept of H-implicative almost distributive lattices, which are a special class of lattices with both implicative and almost distributive properties. This study is aimed at extending the understanding of lattice structures and their ideals, particularly focusing on LI-ideals, a novel concept within H-implicative almost distributive lattices.

Methods

We define an LI-ideal of an implicative almost distributive lattice L and investigate its properties. The paper demonstrates that every LI-ideal in L is an almost distributive lattice ideal of L. Additionally, we explore the relationship between filters and LI-ideals, and we study the process of generating an LI-ideal from a given set. Lastly, we examine the construction of quotient structures via LI-ideals.

Results

We present several examples showing that every almost distributive lattice ideal is also an LI-ideal in an H-implicative almost distributive lattice. The study establishes key relationships between the concepts of filters and LI-ideals. Furthermore, we provide a method for generating an LI-ideal from a set and construct a quotient structure using an LI-ideal.

Conclusions

The paper introduces new concepts and relationships within the study of H-implicative almost distributive lattices. Our findings demonstrate the interconnection between almost distributive lattice ideals and LI-ideals and offer insights into how these ideals can be generated and used to construct quotient structures. This work provides a deeper understanding of lattice theory and opens new avenues for further research in the area of lattice ideals.

Filter H-implicative almost distributive lattice LI-ideals and implicative almost distributive lattice..

The author(s) declared that no grants were involved in supporting this work.

Introduction

In order to research the logical system whose propositional value is given in a lattice, Xu, Y. ⁸ proposed the concept of lattice implication algebras and discussed some of their properties. Xu, Y. and Qin, K. Y. ⁷ discussed the properties of lattice H-implication algebras. Jun, Y. B. and Xu, et al. ^{3,
9} introduced the notions of a filters in a lattice implication algebra and investigate their properties. Jun, Y. B. et al. ⁴ proposed the concept of an LI-ideal of lattice implication algebra. They discussed the relationship between filters and LI-ideals and studied how to generate an LI-ideal by a set. They constructed quotient structure by using an LI-ideal. Kolluru,V. and Bekele, B. ⁵the concept of implicative algebras was introduced, and several key properties were established. It was also proven that every implicative algebra is lattice implication algebra. In 1981, Swamy, U. M., and Rao, G. C. introduced the concept of an almost distributive lattice (ADL) as a common abstraction for many of the existing lattice structure. ⁶

The existing ring-theoretic and lattice-theoretic generalizations of Boolean algebra have been explored in various studies. Berhanu Assaye, Mihret Alamneh, and Tilahun Mekonnen ¹ introduced the concept of implicative almost distributive lattices (IADLs) as a generalization of implicative algebras within the class of almost distributive lattices (ADLs). In this paper, we prove several properties and equivalence conditions within the framework of IADLs.

We also introduced filter, implicative filter in IADL. ² In this paper we discuss H-implicative almost distributive lattice (in short, H-IADL) and give equivalent conditions. We introduce LI-ideals in IADL and show that every LI-ideal is an ADL ideal. We give an example that an ADL ideal may not be an LI-ideal, and show that every ADL ideal is an LI-ideal in H-IADL. We discuss the relationship between filters and LI-ideal, and study how to generate an LI-ideal by a set. We construct quotient structure by using an LI-ideal.

In the following, we give some important definitions and results that will be useful in this study.

Preliminaries

Definition 2.1

⁶ An algebra ( L , ∨ , ∧ , 0 ) of type ( 2 , 2 , 0 ) is called an almost distributive lattice (ADL) with 0 if it satisfies the following axioms: 1.

( x ∨ y ) ∧ z = ( x ∧ z ) ∨ ( y ∧ z )

x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z )

( x ∨ y ) ∧ y = y

( x ∨ y ) ∧ x = x

x ∨ ( x ∧ y ) = x

0 ∧ x = 0 , for all x , y , z ∈ L .

If (L, ∨, ∧, 0) is an ADL, for any x , y ∈ L , define x ≤ y if and only if x = x ∧ y or equivalently x ∨ y = y , then ≤ is a partial ordering on L.

Definition 2.2

⁶ Let L be an ADL. An element m ∈ L is called maximal if for any x ∈ L , m ≤ x implies m = x .

Definition 2.3

⁶ A non-empty subset I of an ADL L is called an ideal of L, if it satisfies the following: i.

x , y ∈ I implies x ∨ y ∈ I .

ii.

x ∈ I and y ∈ L implies that x ∧ y ∈ I . we call I as an ADL ideal of L .

The following important property of the ideals is very useful to develop the algebra of ideals. If I is an ideal of ADL L and a , b ∈ L , then a ∧ b ∈ I if and only if b ∧ a ∈ I .

Definition 2.4

⁶ Let L be an ADL. For any a ∈ L , principal filter of L generated by a is [ a ) = { x ∨ a : x ∈ L }

Definition 2.5

[6] A non-empty subset F of an ADL L is called a filter of L if it satisfies the following: i.

x , y ∈ F implies x ∧ y ∈ F .

ii.

x ∈ F and y ∈ L implies that y ∨ x ∈ F .

Definition 2.6

⁶ A non-empty subset S of an ADL L is called a sub ADL of L if 1.

0 ∈ S

x ∧ y ∈ S and x ∨ y ∈ S , for any x , y ∈ S .

Theorem 2.7

⁶ Let F is filter of an ADL L and x , y ∈ L . Then x ∨ y ∈ F if and only if y ∨ x ∈ F .

Lemma 2.8

⁶ Every ideal (filter) of L is a sub ADL of ADL L.

Definition 2.9

⁵ An algebra ( L , → , ′ , 0 , 1 ) of type ( 2 , 1 , 0 , 0 ) is called implicative algebra if it satisfies the following conditions: 1.

x → ( y → z ) = y → ( x → z )

1 → x = x

x → 1 = 1

x → y = y ′ → x ′

( x → y ) → y = ( y → x ) → x

0 ′ = 1 , for x , y , z ∈ L

Theorem 2.10

⁵ Let ( L , → , ′ , 0 , 1 ) be an implicative algebra. Then ( L , ∨ , ∧ , → , ′ , 0 , 1 ) is a lattice implication algebra.

Definition 2.11

¹ Let ( L , ∨ , ∧ , 0 , m ) be an ADL with 0 and maximal element m. Then an algebra ( L , ∨ , ∧ , → , ′ , 0 , m ) of type ( 2 , 2 , 2 , 1 , 0 , 0 ) is called implicative almost distributive lattice (IADL) if it satisfies the following conditions: 1.

x ∨ y = ( x → y ) → y

x ∧ y = [ ( x → y ) → x ′ ] ′

x → ( y → z ) = y → ( x → z )

m → x = x

x → m = m

x → y = y ′ → x ′

0 ′ = m , for all x , y , z ∈ L .

Now we define the relation ≤ on an IADL L as follows: x ≤ y ⇔ x → y = m , for all x , y ∈ L . The relation ≤ on L is a partial ordering.Thus ( L , ≤ ) is a poset.

Theorem 2.12

¹ In an IADL L, for all x , y , z ∈ L the following conditions hold: 1.

[ ( x → y ) → y ] ∧ m = [ ( y → x ) → x ] ∧ m

[ ( ( x → y ) → x ′ ) ′ ] ∧ m = [ ( ( y → x ) → y ′ ) ′ ] ∧ m

x → x = m

m ′ = 0

( x ′ ) ′ = x

x ′ = x → 0

0 → x = m

x → y = m = y → x implies x = y .

If x → y = m and y → z = m , then x → z = m

10.

x ≤ y if and only if z → x ≤ z → y and y → z ≤ x → z

11.

( ( x → y ) → y ) → y = x → y

12.

( x → y ) → ( ( y → z ) → ( x → z ) ) = m , ( x → y ) → ( ( z → x ) → ( z → y ) ) = m

13.

( x → z ) → ( x → y ) = ( z → x ) → ( z → y ) = ( x ∧ z ) → y .

14.

x → y ≤ ( y → z ) → ( x → z )

15.

( x ∧ y ) ′ = x ′ ∨ y ′ , ( x ∨ y ) ′ = x ′ ∧ y ′

16.

x ≤ y implies y ′ ≤ x ′ .

17.

( x ∨ y ) → z = ( x → z ) ∧ ( y → z )

18.

( x ∧ y ) → z = ( x → z ) ∨ ( y → z )

19.

x → ( y ∧ z ) = ( x → y ) ∧ ( x → z )

20.

x → ( y ∨ z ) = ( x → y ) ∨ ( x → z ) .

Definition 2.13

² Let L be an IADL. 1.

A subset F of L is called a filter of L if it satisfies:

( F ₁) m ∈ F

( F ₂) x ∈ F and x → y ∈ F implies y ∈ F , for all x , y ∈ L .

A subset F of L is called implicative filter of L if it satisfies

( F ₁) m ∈ F

( I) x → y ∈ F and x → ( y → z ) ∈ F implies x → z ∈ F , for all x , y , z ∈ L .

Lemma 2.14

² Let F be a non-empty subset of an IADL L. Then F is a filter of L if and only if it satisfies for all x , y ∈ F and z ∈ L :

x ≤ y → z implies z ∈ F

Lemma 2.15

² Every filter F of an IADL L has the following property: x ≤ y and x ∈ F implies y ∈ F .

Definition 2.16

⁷ A lattice implication algebra L is called a lattice H-implication algebra, if for any x , y , z ∈ L , x ∨ y ∨ ( ( x ∧ y ) → z ) = 1 .

Theorem 2.17

⁷ Let L be a lattice H-implication algebra. Then for any x , y , z ∈ L , x → ( y → z ) = ( x → y ) → ( x → z ) .

Theorem 2.18

¹ Let L be an IADL, then for any x , y , z ∈ L , ( x → ( y → z ) ) → ( ( x → y ) → ( x → z ) ) = x ∨ y ∨ ( ( x ∧ y ) → z ) .

Definition 2.19

⁴ Let L be a lattice implication algebra. An LI-ideal A of L is a non-empty subset of L such that 1.

0 ∈ A ,

y ∈ A and ( x → y ) ′ ∈ A implies x ∈ A , for x , y ∈ L .

Theorem 2.20

^{4,
9} Let L be a lattice implication algebra. Every LI-ideal of L is a lattice ideal.

Theorem 2.21

⁴ In lattice H-implication algebra, every lattice ideal is an LI-ideal.

Definition 2.22

⁶ An ADL ( L , ∨ , ∧ ) is said to be associative if the operation ∨ in L is associative.

Definition 2.23

⁶ An equivalence relation θ on an ADL L is called a congruence relation on L if for all a , b , c , d ∈ L , ( a , b ) , ( c , d ) ∈ θ implies ( a ∧ c , b ∧ d ) ; ( a ∨ c , b ∨ d ) ∈ θ .

LI-ideals in implicative almost distributive lattices

In this section first we introduce H-implicative almost distributive lattice (H-IADL). Then we introduce LI-ideals in IADLs and discuss some of their properties. Finally, we give some characterizations of LI-ideals.

H-implicative almost distributive lattices

In this subsection we introduce H-implicative almost distributive lattice (H-IADL) and study the conditions of IADL being H-IADL. Definition 3.1

An implicative almost distributive lattice (IADL) L is called H-implicative almost distributive lattice (H-IADL) if it satisfies: x ∨ y ∨ ( ( x ∧ y ) → z ) = m , for all x , y , z ∈ L

Example 3.2

Let L = { 0 , x , y , m } be a set. Define the partially ordered relation on L as ≤ = { ( 0 , x ) , ( 0 , m ) , ( 0 , y ) , ( x , m ) , ( y , m ) , ( 0 , 0 ) , ( x , x ) , ( y , y ) , ( m , m ) } and also define x ∧ y = min { x , y } , x ∨ y = max { x , y } for all x , y , z ∈ L . Define the unary operation ^′ and binary operation → as shown in the tables below respectively. Table 1.

a a

0 m

x y

y x

m 0

Table 2.

→ 0 x y m

0 m m m m

x y m y m

y x x m m

m 0 x y m

Then clearly ( L , ∨ , ∧ , → , ′ , 0 , m ) is an IADL and hence it can be easily verified that L is an H-IADL.

In the following some properties of H-IADLs are discussed.

Theorem 3.3

Let L be an H-IADL. Then x → ( y → z ) = ( x → y ) → ( x → z ) for any x , y , z ∈ L .

Proof.

Let L be an H-IADL and x , y , z ∈ L . Now ( x → ( y → z ) ) → ( ( x → y ) → ( x → z ) ) = ( ( x → ( y → z ) ) → ( ( y ′ → x ′ ) → ( z ′ → x ′ ) ) = ( y ′ → x ′ ) → ( ( y → ( x → z ) ) → ( x → z ) ) = ( x → y ) → ( y ∨ ( x → z ) ) = ( x → ( y → y ) ) ∨ ( ( x → y ) → ( x → z ) ) = x ∨ y ∨ ( ( x ∧ y ) z ) = m (by Definition 3.1 and Theorem 2.12). Therefore ( x → ( y → z ) ) ≤ ( ( x → y ) → ( x → z ) ) . (1).

On the other hand, ( ( x → y ) → ( x → z ) ) → ( x → ( y → z ) ) = y → ( ( x → y ) → ( x → z ) ) → ( x → z ) ) = y → ( ( x → y ) ∨ ( x → z ) ) = ( y → ( x → y ) ) ∨ ( y → ( x → z ) ) = m ∨ ( y → ( x → z ) ) = m .

Therefore, ( ( x → y ) → ( x → z ) ) ≤ ( x → ( y → z ) ) … . ( 2 ) . Hence by (1) and (2), we get x → ( y → z ) = ( x → y ) → ( x → z ) . □

Corollary 3.4

Let L be an IADL. Then the following are equivalent: 1.

L is an H-IADL;

For any x , y ∈ L , x → ( x → y ) = x → y

For any x , y , z ∈ L , x → ( y → z ) = ( x → y ) → ( x → z )

For any x , y , z ∈ L , x → ( y → z ) = ( x ∧ y ) → z

For any x , y , z ∈ L , ( x → ( y → z ) ) → ( ( x → y ) → ( x → z ) ) = m .

Proof.

Let L be an IADL and x , y , z ∈ L .

( 1 ) ⇒ ( 2 ) . Suppose L is an H-IADL. Then using Theorem 3.3, we have x → ( x → y ) = ( x → x ) → ( x → y ) = m → ( x → y ) = x → y .

( 1 ) ⇒ ( 4 ) . Suppose x → ( x → y ) = x → y . By assumption and Theorem 2.12, ( x ∧ y ) → z = ( x ∧ y ) → [ ( x ∧ y ) → z ] = [ ( x ∧ y ) → ( x → z ) ] ∨ [ ( x ∧ y ) → ( y → z ) ] = [ ( x ∧ y ) → ( x → z ) ] ∨ [ ( x ∧ y ) → ( y → z ) = [ ( x → ( x → z ) ) ∨ ( ( y → ( x → z ) ) ] ∨ [ ( x → ( y → z ) ] ∨ [ ( y → ( y → z ) = [ ( x → z ) ∨ ( y → ( x → z ) ) ] ∨ [ ( x → ( y → z ) ∨ ( y → z ) ] = [ ( x → z ) → ( y → ( x → z ) ) → ( y → ( x → z ) ] ∨ [ ( ( x → ( y → z ) ) → ( y → z ) ) → ( y → z ) ] = [ y → ( ( x → z ) → ( x → z ) ) ] ∨ [ ( x → ( y → z ) ) ] = [ y → m ) → ( y → ( x → z ) ) ∨ [ x → ( y → z ) ] = [ m → ( y → ( x → z ) ] ∨ [ x → ( y → z ) ] = [ y → ( x → z ) ] ∨ [ x → ( y → z ) ] = [ x → ( y → z ) ] ∨ [ x → ( y → z ) ] = x → ( y → z ) .

(4) ⇒ (3). Suppose x → ( y → z ) = ( x ∧ y ) → z . Now ( x ∧ y ) → z = [ ( x → y ) → x ′ ] ′ → z = z ′ → [ ( x → y ) → x ′ ] = ( x → y ) → ( z ′ → x ′ ) = ( x → y ) → ( x → z ) = x → ( y → z ) .

( 3 ) ⇒ ( 4 ) . Suppose x → ( y → z ) = ( x → y ) → ( x → z ) . Now ( x ∧ y ) → z = [ ( x → y ) → x ′ ] ′ → z = z ′ → ( ( x → y ) → x ′ ) = ( x → y ) → ( x → z ) = x → ( y → z ) (by Theorem 3.3).

( 3 ) ⇒ ( 5 ) . Suppose x → ( y → z ) = ( x ∧ y ) → z . Now ( x → ( y → z ) ) → ( ( x → y ) → ( x → z ) ) = ( ( x ∧ y ) → z ) → ( ( x → y ) → ( x → z ) ) = ( ( x → y ) → ( x → z ) ) → ( ( x → y ) → ( x → z ) ) = m . Therefore [ x → ( y → z ) ] → [ ( x → y ) → ( x → z ) ] = m . Hence (5) holds.

( 3 ) ⇒ ( 1 ) . Suppose ( x → ( y → z ) ) → [ ( x → y ) → ( x → z ) ] . Then using Theorem 2.12 and Definition 3.1, ( x → ( y → z ) ) → ( ( x → y ) → ( x → z ) ) = ( x → y ) → ( y → ( ( x → z ) → ( x → z ) ) = ( x → y ) → ( y ∨ ( x → z ) ) = ( ( x → y ) → y ) ∨ ( ( x → y ) → ( x → z ) ) = x ∨ y ∨ ( ( x → y ) → ( x → z ) ) = x ∨ y ∨ ( ( x ∧ y ) → z ) = m . Therefore, L is an H-IADL. □ Definition 3.5

An IADL ( L , ∨ , ∧ , → , ′ , 0 , m ) is said to be associative IADL if ∨ in L is assocative

Theorem 3.6

Let L be an associative IADL. Then the following statements are equivalent. 1.

L is an H-IADL;

for any x , y ∈ L and z ∈ [ x , m ] , z → ( x → y ) = x → y ;

for any x , y ∈ L , ( x → y ) → x = x .

for any x ∈ L , x ∨ x ′ = m

Proof.

Let L be an IADL and x , y , z ∈ L .

( 1 ) ⇒ ( 2 ). Suppose L is an H-IADL. Let z ∈ [ x, m]. Then we have x ≤ z . From Theorem 2.12, we have z → ( x → y ) ≤ x → ( x → y ) . This implies again x → y ≤ z → ( x → y ) ≤ x → ( x → y ) = x → y (by Theorem 2.12 and Corollary 3.4), i.e., z → ( x → y ) = x → y . Therefore (2) holds.

( 2 ) ⇒ ( 1 ) . Putting z = x, we have x → ( x → y) = x → y. Then it follows from Corollary 3.4 that L is an H-IADL.

( 3 ) → ( 1 ) . Suppose ( x → y ) → x = x . Now ( x → ( x → y ) ) → ( x → y ) = ( ( ( x → y ) → x ) → x ) = x → x = m . It follows that x → ( x → y ) ≤ x → y but we know from Theorem 2.12 that x → y ≤ x → ( x → y ) . Therefore x → y = x → ( x → y ) , it follows from Corollary 3.4 that L is an H-IADL.

( 1 ) ⇒ ( 3 ) . Suppose L is an H-IADL. Using Corollary 3.4 and Theorem 2.12, we have (( x → y) → x) → x = ( x → ( x → y)) → ( x → y) = ( x → y) → ( x → y) = m. Therefore ( x → y ) → x ≤ x and clearly x → ( ( x → y ) → x ) = m . This implies x → ( x → y ) ≤ x . Hence ( x → y ) → x = x .

( 3 ) ⇒ ( 4 ) . Suppose ( x → y ) → x = x . Now x ′ = ( x ′ → 0 ) → x ′ = x → x ′ (by supposition), it follows x ∨ x ′ = ( x → x ′ ) → x ′ = x ′ → x ′ = m . That is (4) holds.

( 4 ) ⇒ ( 3 ) . Suppose x ∨ x ′ = m . Then ( x ∨ x ′ ) ′ = x ′ ∧ x = m ′ = 0 . Therefore L is complemented IADL. Hence ∨ is associative. Now using Theorem 2.12 and our assumption we get ( x → y ) → ( x ′ ∨ y ) = ( ( x → y ) → x ′ ) ∨ ( ( x → y ) → y ) = y ′ ∨ x ′ ∨ x ∨ y = m and ( x ′ ∨ y ) → ( x → y ) = ( ( ( x → 0 ) → y ) → y ) → ( x → y ) = x → ( ( ( ( x → 0 ) → y ) → y ) → y ) = x → ( ( x → 0 ) → y ) = x → ( ( y → 0 ) → x ) = y ′ → ( x → x ) = y ′ → m = m (by Theorem 3.3). Hence x → y = x ′ ∨ y . It follows that ( x → y ) → x = ( x ′ ∨ y ) → x = ( x ′ ∨ y ) ′ ∨ x = ( x ∧ y ′ ) ∨ x = ( ( x ∧ y ′ ) → x ) → x = [ ( x → x ) ∨ ( y ′ → x ) ] → x = m → x = x , that is (3) holds. □

Corollary 3.7

Let L be an H-IADL. Then for any x , y ∈ L , y ∨ ( x → y ) ′ = x ∨ y .

Proof.

Let L be an H-IADL and x , y ∈ L . Now ( y ∨ ( x → y ) ′ ) → ( x ∨ y ) = ( y ∨ ( x → y ) ′ ) → [ ( x → y ) → y ] = ( x → y ) → [ ( y ∨ ( x → y ) ′ ) → y ) ] = ( x → y ) → [ ( y → y ) ∧ ( ( x → y ) ′ → y ) ] = ( x → y ) → [ ( x → y ) ′ → y ] = y ′ → [ ( x → y ) → ( x → y ) ] = m . This implies y ∨ ( x → y ) ′ → ( x ∨ y ) = m . Therefore, y ∨ ( x → y ) ′ ≤ x ∨ y …(1). Conversely ( x ∨ y ) → [ y ∨ ( x → y ) ′ ] = ( ( x ∨ y ) → y ) ∨ [ ( x ∨ y ) → ( x → y ) ′ ] = ( x → y ) ∨ [ ( x → y ) → ( x ′ ∧ y ′ ) ] = ( x → y ) ∨ [ ( ( x → y ) → x ′ ) ∧ ( ( x → y ) → y ′ ) ] = [ ( x → y ) ∨ ( ( x → y ) → x ′ ] ∧ [ ( x → y ) ∨ ( ( x → y ) → y ′ ) ] = [ ( x → y ) → ( ( x → y ) → x ′ ) → ( ( x → y ) → x ′ ) ] ∧ [ ( x → y ) → ( ( x → y ) → y ′ ) → ( ( x → y ) → y ′ ) ] = [ ( x → y ) → x ′ ) → ( ( x → y ) → x ′ ) ] ∧ [ ( ( x → y ) → y ′ ) → ( ( x → y ) → y ′ ) ] = m ∧ m = m . This implies x ∨ y ≤ y ∨ ( x → y ) ′ .... (2). Hence by (1) and (2), we have y ∨ ( x → y ) ′ = x ∨ y . □

LI-ideals in implicative almost distributive lattices

In this section we introduce LI-ideal in IADL. We discuss some characterization of LI-ideal with that of H-IADL. We observe relation between LI-ideal and filters in IADL, construction of LI-ideal using a set and lastly we discuss quotient structure of LI-ideal in IADL. Definition 3.8

Let L be an IADL. An LI-ideal A of L is a subset of L such that

( I ₁) 0 ∈ A and

( I ₂) y ∈ A and ( x → y ) ′ ∈ A implies that x ∈ A .

In this case if A ≠ L ,then A is proper LI-ideal of L. A proper LI-ideal A of L is said to be maximal if it is not properly contained in any other proper LI-ideal of L. That is, if I is any other LI-ideal of L such that A ⊆ I ⊆ L implies either A = I or I = L , then A is maximal LI-ideal of L.

Remark 3.9

From the above definition we observe that {0} and L are trivial examples of LI-ideals.

The following example shows that there is a proper LI-ideal in IADL. Example 3.10

Let L = { 0 , x , y , z , w , m } be the underlining set with partial ordering ≤ = { ( 0 , w ) , ( 0 , x ) , ( 0 , m ) , ( 0 , z ) , ( 0 , y ) , ( w , x ) , ( w , m ) , ( w , y ) , ( z , y ) , ( z , m ) , ( y , m ) , ( 0 , 0 ) , ( w , w ) , ( x , x ) , ( z , z ) ( y , y ) , ( m , m ) } . Define the unary operation ′ and binary operation → as shown in the tables below respectively. Table 3.

a a

0 m

x z

y w

z x

w y

m 0

Table 4.

–→ 0 x y z w m

0 m m m m m m

x z m y z y m

y w x m y x m

z x x m m x m

w y m m y m m

m 0 x y z w m

Define the binary operation ∨ and ∧ by x ∨ y = ( x → y ) → y ∧ y = [ ( x → y ) → x ′ ] ′ for all x , y ∈ L . Then clearly ( L , ∨ , ∧ , → , ′ , 0 , m ) is an IADL. We can easily verify that A = { 0 , z } is an LI-ideal of L.

Theorem 3.11

Let A be an LI-ideal of IADL L. If x ∈ A and y ≤ x for some ∈ L , then y ∈ A .

Proof.

Let A be an LI-ideal of IADL L. Suppose x ∈ A and y ≤ x for some y ∈ L .

Claim: y ∈ A for some y ∈ L . Since y ≤ x implies ( y → x ) ′ = m ′ = 0 ∈ A then by definition of LI-ideal ( I ₂) we have y ∈ A . □

Theorem 3.12

Let A be a non-empty subset of IADL L. Then A is an LI-ideal of L if and only if it satisfies for all x , y ∈ A and z ∈ L , ( z → x ) ′ ≤ y implies z ∈ A .

Proof.

Suppose that A is an LI-ideal and x , y ∈ A , z ∈ L . If ( z → x ) ′ ≤ y , then ( z → x ) ′ ∈ A by Theorem 3.11. Using definition 3.8 ( I ₂) we obtain z ∈ A . Conversely, suppose that for all x , y ∈ A and z ∈ L , ( z → x ) ′ ≤ y implies z ∈ A . Since A is a non empty subset of L, we assume x ∈ A . Because ( 0 → x ) ′ ≤ x , we have 0 ∈ A , and so I ₁ holds for A. Let ( x → y ) ′ ∈ A and y ∈ A . Since ( x → y ) ′ ≤ ( x → y ) ′, we have x ∈ A , and so ( I ₂) holds for A. Hence, A is an LI-ideal of L. □

Theorem 3.13

Let L be an IADL. Every LI-ideal of L is an ADL ideal of L.

Proof.

Let A be an LI-ideal of IADL L. i.

Let x , y ∈ A . Claim: x ∨ y ∈ A .

Now ( ( x ∨ y ) → y ) ′ = ( ( ( x → y ) → y ) → y ) ′ = ( x → y ) ′ ≤ ( x ′ ) ′ = x (since ( x → y ) ′ → x = x ′ → ( x → y ) = x → ( x ′ → y ) = y ′ → ( x → x ) = m ) . This implies ( ( x ∨ y ) → y ) ′ ∈ A (by Theorem 3.11). Thus x ∨ y ∈ A (by Definition 3.8 ( I ₂)).

ii.

Suppose a ∈ A and x ∈ L .

Claim: a ∧ x ∈ A .

Now ( ( a ∧ x ) → a ) ′ = ( ( a → a ) ∨ ( x → a ) ) ′ = ( m ∨ ( x → a ) ) ′ = m ′ = 0 ∈ A . This implies a ∧ x ∈ A . Hence, from (i) and (ii) A is an ADL ideal of L. □

Remark 3.14

The converse of Theorem 3.13 is not true. Consider Example 3.10. The set A = {0 , w} is an ADL ideal. But it is not an LI-ideal since ( x → w ) ′ = w ∈ A and ∉ A . □

Now we have the following characterization. Theorem 3.15

In H-IADL L, every ADL ideal is an LI-ideal.

Proof.

Let x , y ∈ L and A be an ADL ideal of H-IADL L. Assume that y ∈ A and ( x → y ) ′ ∈ A . We want to show that x ∈ A . Clearly 0 ∈ A . From Corollary 3.7, we have x ∨ y = y ∨ ( x → y ) ^′. Then it follows from Definition 2.3 (ii) that x ∨ y = y ∨ ( x → y ) ′ ∈ A . Since x = ( x ∨ y ) ∧ x ∈ A , then we have x ∈ A . Hence A is an LI-ideal of L. □

Lemma 3.16

Let A be a non empty subset of H-IADL L. Then A is an LI-ideal of L if and only if for every x , y ∈ L , x , y ∈ A if and only if x ∨ y ∈ A .

Proof.

Suppose A is an LI-ideal of H-IADL L. Let x , y ∈ A . Since ( ( x ∨ y ) → y ) ′ = ( ( x → y ) ∧ ( y → y ) ) ′ = 0 ∈ A and A is an LI-ideal, then we have x ∨ y ∈ A . Also ( x → ( x ∨ y ) ) ′ = m ′ = 0 ∈ A implies x ∈ A as A is an LI-ideal. Conversely suppose x , y ∈ A if and only if x ∨ y ∈ A and ( x → y ) ′ ∈ A trivially x ∈ A . Hence, A is an LI-ideal.

For any A and B of IADL L we set A ∧ B = { a ∧ b| a ∈ A and b ∈ B}.

Theorem 3.17

If A and B are LI-ideals of an H-IADL of L, then so is A ∧ B .

Proof.

Let A and B be two LI-ideals of H-IADL L. Let x , y ∈ A ∧ B . Then x = a ₁ ∧ b ₁ for a ₁ ∈ A and b ₁ ∈ B and y = a ₂ ∧ b ₂ for a ₂ ∈ A and b ₂ ∈ B. Now x ∨ y = ( a ₁ ∧ b ₁) ∨ ( a ₂ ∧ b ₂) = (( a ₁ ∧ b ₁) ∨ a ₂) ∧ (( a ₁ ∧ b ₁) ∨ b ₂) ∈ A ∧ B as (( a ₁ ∧ b ₁) ∨ a ₂ ∈ A and ( ( a 1 ∧ b 1 ) ∨ b 2 ) ∈ B . Thus , x ∨ y ∈ A ∧ B . Conversely assume x ∨ y ∈ A ∧ B for all x , y ∈ L . Then x ∨ y = a ∧ b for some a ∈ A and b ∈ B . We observe that x = x ∧ ( x ∨ y ) = x ∧ ( a ∧ b ) ≤ a ∈ A implies x ∈ A . Similarly one can show that x ∈ B and that y ∈ A and y ∈ B . Thus x , y ∈ A ∧ B . Hence, A ∧ B is an LI-ideal of L. □

Theorem 3.18

If A and B are LI-ideals of an H-IADL L, then A ∧ B = A ∩ B .

Proof.

Let x ∈ A ∧ B . Then x = a ∧ b for some a ∈ A and b ∈ B . Since x = b ∧ a ≤ a ∈ A implies a ∧ b ∈ A (as A is an ADL ideal by Theorem 3.15 and properties of ideal in ADL) and x = a ∧ b ≤ b ∈ B , then from Theorem 3.11, x ∈ A and also x ∈ B and hence x ∈ A ∩ B . Therefore, A ∧ B ⊆ A ∩ B . Conversely suppose x ∈ A ∩ B then x = x ∧ x ∈ A ∧ B . Therefore, A ∩ B ⊆ A ∧ B . Hence, A ∧ B = A ∩ B . □

Theorem 3.19

If A is an LI-ideal of an H-IADL L and a ∈ L , then the set K = { x ∈ L | x ′ → a ∈ A } is an LI-idel of L.

Proof

Let x , y ∈ K . Then x ′ → a ∈ A and y ′ → a ∈ A . Using Theorem 2.13 that ( x ∨ y ) ′ → a = ( x ′ ∧ y ′ ) → a = ( x ′ → a ) ∨ ( y ′ → a ) ∈ A (by Theorem 3.24) so that x ∨ y ∈ K . Conversely let x , y ∈ L be such that x ∨ y ∈ K . Then ( x ∨ y ) ′ → a ∈ A . Using Theorem 2.12 and Theorem 3.3, we have ( x ∨ y ) ′ → a = a ′ → ( x ∨ y ) ) = a ′ → ( ( x → y ) → y ) = ( a ′ → ( x → y ) ) → ( a ′ → y ) = ( a ′ → x ) → ( a ′ → y ) → ( a ′ → y ) = ( a ′ → x ) ∨ ( a ′ → y ) = ( x ′ → a ) ∨ ( y ′ → a ) . Thus ( x ′ → a ) ∨ ( y ′ → a ) ∈ A which implies that x ′ → a ∈ A and y ′ → a ∈ A because A is an LI-ideal of H-IADL and Lemma 3.16. This implies that x ∈ K and y ∈ K . Hence by Theorem 2.13, K is an LI-ideal of L.

Proposition 3.20

Let L be an H-IADL and a ∈ L .Then there is no proper LI-ideal of L containing a and a ′ simultaneously.

Proof.

Let A be a proper LI-ideal of H-IADL L containing a and a′ simultaneously. Then m = a ∨ a ′ ∈ A , and hence A = L a contradiction. Therefore, A is not a proper LI-ideal of H-IADL L containing a and a′ simultaneously. □

For any non-empty subset A of an IADL L, we define A ′ = { x ′ | x ∈ A } . It is obvious that every non empty subset A of IADL L is not a filter. Similarly the set A ^′ for every subset A of L is not an LI-ideal in general. In fact the dual concept of a filter is one of an LI-ideal in IADL. Then we have the following theorem to verify this ideal. Theorem 3.21

Let A be a non empty subset of IADL L. Then A is a filter of L if and only if A′ is an LI-ideal of L.

Proof.

Assume that A is a filter of IADL L. Then m ∈ A , and so m ′ = 0 ∈ A ′ . Let ( x → y ) ′ ∈ A ′ and y ∈ A ′ for all x , y ∈ L . Then ( x → y ) ′ = u ′ and y = v ′ for some u , v ∈ A . Thus v → x ′ = x → v ′ = x → y = ( ( x → y ) ′ ) ′ = ( u ′ ) ′ = u ∈ A . Since A is a filter, we have x ′ ∈ A and so x = ( x ′ ) ′ ∈ A ′ . This proves that A′ is an LI-ideal of L. Conversely let x , y ∈ L be such that x ∈ A and x → y ∈ A . We want to show y ∈ A . Suppose that A′ is an LI-ideal of L. Then x ′ ∈ A ′ and ( y ′ → x ′ ) ′ = ( x → y ) ′ ∈ A ′. As A′ is an LI-ideal, it follows from Definition 3.8 ( I ₂) that y ′ ∈ A ′ or y ∈ A . Also 0 ′ = m ∈ A . Hence, A is a filter of L. □

Theorem 3.22

Suppose { B _i : i ∈ J } for index set J is a non-empty family of LI-ideal of IADL L. Then A = ∩ i _{∈
J} B _i is also an LI-ideal of L, for any i ∈ J of index set J.

Proof.

Let A = { B _i : B _i is an LI − ideal of L } .

We need to show A = ∩ _i∈J B _i is also an LI-ideal of L. 1.

Clearly 0 ∈ B _i for each ∈ J . This implies 0 ∈ ∩ _{i∈
J} B _i

Since ( x → y ) ′ ∈ B _i and y ∈ B _i implies x ∈ B _i for each i ∈ J , then ( x → y ) ′ ∈ ∩ i ∈ J Bi and y ∈ ∩ _{i∈
J} B _i implies x ∈ ∩ _{i∈
J} B _i.

Therefore, A = ∩ _{i∈
J} B _i is an LI-ideal of L.□

Let A be a subset of an IADL L. Then the least LI-ideal containing A is called the LI-ideal generated by A, written ⟨ A ⟩ . From Remark 3.14, L is clearly an LI- ideal Containing A. If A = { a } , ⟨ { a } ⟩ is written ⟨ a ⟩ . In short we shall write [ a ₁, a ₂, a ₃, · · ·, a _n, x] for a ₁ → ( a ₂ → (· · · → ( a _n → x) · · ·), and write [ a ⁿ, x] if a ₁ = a ₂ = · · · = a _n = a. So in our case we define ⟨ A ⟩ = { x ∈ L | a n ′ → ( · · ( a 1 ′ → x ′ ) ⋯ ) = m for some a ₁, · · ·, a _n ∈ A} and ⟨ a ⟩ = { x ∈ L [ ( a ′ ) ⁿ , x ′ ] = m for some n ∈ N } .

For any natural number n we define n( x) → y recursively as follows: 1( x) → y = x → y and ( n + 1)( x) → y = x → ( n( x) → y).

Using definition 2.11(3) repeatedly, we know that the following identity in IADL holds: i.

z → ( y ₁ → ( y ₂ → (· · · ( y _n → x) · · ·))) = y ₁ → ( y ₂ → (· · · ( y _n → ( z → x) · · ·)). As a special case of (i) we have

ii.

z → ( n ( y ) → x ) = n ( y ) → ( z → x ) . □

In the following theorem we can describe elements of ⟨ A ⟩ . Theorem 3.23

If A is a non-empty subset of an IADL L, then ⟨ A ⟩ = {x ∈ L| a ₁ ^′→ ⋯ ( a ′ 1 → x ^′) · · ·) = m for some a ₁, · · ·, a _n ∈ A } is the least LI-idea of L containing.

Proof.

Let us denote U = { x ∈ L | ( a ′ n → (· · · ( a ₁′ → x′) · · ·) = m for some a ₁ , · · ·, a _n ∈ A }. We claim U = ⟨ A ⟩ . That is, we need to prove 1) ⊆ U , 2) U is an LI-ideal, 3) U ⊆ V for any LI-ideal V containing A. To prove (1). Since A is non-empty there exists a ∈ A such a ′ → a ′ = m . This is implies a ∈ ¹U. Therefore, ⊆ U . To prove 2) For a ∈ A , a ′ → 0 ′ = m implies 0 ∈ U . Let ( x → y ) ′ ∈ U and y ∈ U , then there exists a _i ∈ A ( i = 1 , 2 , 3 , ⋯ , n ) and b _j ∈ A ( j = 1 , 2 , 3 , · · ·, p) such that a′ → ( · · · ( a′ → (( x → y)′) ^′ · · ·) = m…( ∗ ) .

b p ′ → ( · · · ( b 1 ′ → y′) · · ·) = m…( ∗ ∗ ) . From ( ∗ ) , we have a n ′ → (· · · (a′ →(( y′ → x′) · · ·) = m … ( ∗ ∗ ∗ ) . This implies y′ ≤ a n ′ → ( · · ·( a1 ^′→ x′) · · ·) … ( ∗ ∗ ∗ ∗ )

Combining ( ∗ ∗ ) , ( ∗ ∗ ∗ ∗ ) and properties of IADL, we get m = b p ′ → ( · · · ( b 1 ′ →y′) · · ·) ≤ b p ′ → ( · · ·( b 1 ′ →→ ( · · ·( a 1 ′ → x′) · · ·))) · · ·) and hence b p ′ →(· · · ( b 1 ′ →( a n ′ → ( ·· · ( a′→ x′) · · ·))) · · ·) = m. This shows that ∈ U .

Therefore, U is an LI-ideal of L containing A. To prove 3) Let V be an LI- ideal containing A and let ∈ U . Then ( a n ′ → ( · · ·( a 1 ′ → x′) · · ·) = m for some a ₁ , ⋯ , a _n ∈ A . Thus m = ( a n ′ →(( a n − 1 ′ → ( · · · ( a 1 ′ → x′) · · ·))= ( a n ′ →(( a n − 1 ′ → ( · · · ( a 1 ′ → x′) · · ·))) = ( a n − 1 ′ → a 1 ′ → ( · · a 1 ′ → x ′ ) · ·)) →a _n , which implies that (( a n − 1 ′ → ( · · · ( a 1 ′ → x′) · · ·)) ^′ → a _n )′ = m ^′ = 0 ∈ V .

Since a _n ∈ A ⊆ V and V is an LI-ideal of L, we have ( a n − 1 ′ → ( · · · ( a 1 ′ →x′) · · · ) ) ′ ) ′ ∈ V . Now ( a n − 1 ′ → ( · · · ( a 1 ′ → x′) · · ·))′ = ( a n − 2 ′ → ( · · · ( a 1 ′ → x′) → a _n−1) · · ·))′, since a n − 1 ∈ A ⊆ V , it follows from Definition 3.16 I ₂ that ( a n − 2 ′ → ( · · · ( a′→ x′) · · · ) ) ′ ∈ V . Repeating the process we conclude that ( x → a ) ′ = m ′ = 0 ∈ V and ∈ V . As V is an LI-ideal of L we have ∈ V . This proves that U ⊆ V , hence U = ⟨ A ⟩ . That is, U is the least LI-ideal containing A. □

The following corollary is immediate from Theorem 3.23.

Corollary 3.24

For any element a of an IADL L, we have ⟨ a ⟩ = { x ∈ L : n ( a ′ ) → x ′ = m for some natural number n}.

Theorem 3.25

Let A be an an LI-ideal of IADL L and a ∈ L . Then ⟨ A ∪ { a } ⟩ = { x ∈ L | ( n ( a ′ ) → x ′ ) ′ ∈ A for some n ∈ N } .

Proof.

Let A be an LI-ideal of IADL L and a ∈ L . Consider A _a = { x ∈ L | ( n ( a ′ ) → x ′ ) ′ ∈ A for some n ∈ N } . Since ( n ( a ′ ) → 0 ′ ) ′ = 0 ∈ A , then 0 ∈ A _a. Let ( y → x ) ′ ∈ A _a and x ∈ A _a. Then there exist m , n ∈ N such that ( n ( a ′ ) → ( ( y → x ) ′ ) ′ ) ′ ∈ A and ( m ( a ′ ) → x ′ ) ′ ∈ A . It follows that ( n ( a ′ ) → ( y → x ) ) ′ = u and ( m ( a ′ ) → x ′ ) ′ = v for some u , v ∈ A , so that u ′ = n ( a ′ ) → ( y → x ) and v ′ = m ( a ′ ) → x ′ . Then m = u ′ → u ′ = u ′ → ( n ( a ′ ) → ( y → x ) ) = u ′ → ( n ( a ′ ) → ( x ′ → y ′ ) ) = u ′ → ( x ′ → ( n ( a ′ ) → y ′ ) ) = x ′ → ( u ′ → ( n ( a ′ ) → y ′ ) ) , i . e . , x ′ ≤ u ′ → ( n ( a ′ ) → y ′ ) . Using Theorem 2.12 v ′ = m ( a ′ ) → x ′ ≤ m ( a ′ ) → ( u ′ → ( n ( a ′ ) → y ′ ) ) = u ′ → ( m ( a ′ ) → ( n ( a ′ ) → y ′ ) ) = u ′ → ( ( m + n ) ( a ′ ) → y ′ ) , which implies that v ′ → ( u ′ → ( ( ( m + n ) ( a ′ ) → y ′ ) ′ ) ′ ) = v ′ → ( u ′ → ( ( m + n ) ( a ′ ) → y ′ ) ) = m . Since u , v ∈ A , it follows from Theorem 3.23 that ( ( m + n ) ( a ′ ) → y ′ ) ′ ∈ ⟨ A ⟩ = A . Hence y ∈ Aa and A _a is LI-ideal of L. Since ( n ( a ′ ) → a ′ ) ′ = m ′ = 0 ∈ A , then a ∈ A _a. Let x ∈ A . Since x ′ ≤ a ′ → x ′ = ( ( a ′ → x ′ ) ′ ) ′ , it follows from Theorem 3.19 that ( a ′ → x ′ ) ′ ∈ A i.e., x ∈ A _a and so A ⊆ A _a. Thus A ∪ { a } ⊆ A _a. Finally we show that A _a is the least LI-ideal containing A and a. Let B be any LI-ideal containing A and a, and let x ∈ A _a. Then ( n ( a ′ ) → x ′ ) ′ ∈ A ⊆ B for some n ∈ N , and hence ( ( ( n − 1 ) ( a ′ ) → x ′ ) ) ′ → a ) ′ = ( a ′ → ( ( n − 1 ) ( a ′ ) → x ′ ) ) ′ = ( n ( a ′ ) → x ′ ) ′ ∈ B . Since a ∈ B it follows from definition of LI-ideal we have ( n − 1 ) ( a ′ ) → x ′ ) ′ ∈ B . Repeating this process we obtain x = ( x ′ ) ′ ∈ B . Therefore, A _a is the least LI- ideal containing A _a and a, i.e., ⟨ A ∪ { a } ⟩ = A _a.

Lemma 3.26

Let L be an IADL and a , b , x ∈ L .If n ( a ) → x = m and m ( b ) → x = m for some n , m ∈ N , then there exists k ∈ N such that k ( a ∨ b ) → x = m

Theorem 3.27

Let A be an LI-ideal of an IADL L. Then ⟨ A ∪ { a } ⟩ ∩ ⟨ A ∪ { b } ⟩ = ⟨ A ∪ { a ∧ b } ⟩ for all a , b ∈ L .

Proof.

Let x ∈ ⟨ A ∪ { a } ⟩ ∩ ⟨ A ∪ { b } ⟩ . By Theorem 3.23, there are m , n ∈ N such that ( n ( a ′ ) → x ′ ) ′ ∈ A and ( m ( b ′ ) → x ′ ) ′ ∈ A . Hence (n(a′) → x′)′ = u and (m(b′) → x′)′ = v for some u , v ∈ A . It follows that n ( a ′ ) → x ′ = u ′ and m ( b ′ ) → x ′ = v ′ so that m = v ′ → ( u ′ → ( n ( a ′ ) → x ′ ) ) = m ( a ′ ) → ( v ′ → ( u ′ → x ′ ) ) and m = u ′ → ( v ′ → ( m ( b ′ ) → x ′ ) ) = m ( b ′ ) → ( v ′ → ( u ′ → x ′ ) ) . Using Lemma 3.26, there exists k ∈ N such that k ( a ′ ∨ b ′ ) → ( v ′ → ( u ′ → x ′ ) ) = m . Since a ′ ∨ b ′ = ( a ∧ b ) ′ , we have m = k ( a ′ ∨ b ′ ) → ( v ′ → ( u ′ → x ′ ) ) = k ( a ∧ b ) ′ → ( v ′ → ( u ′ → x ′ ) ) = v ′ → ( u ′ → ( k ( ( a ∧ b ) ′ → x ′ ) ) . Applying Theorem 3.21 we get ( k ( ( a ∧ b ) ′ ) ′ ) ′ ∈ ⟨ A ⟩ = A and hence x ∈ ⟨ A ∪ { a ∧ b } ⟩ by Theorem 3.23. Thus, ⟨ A ∪ { a } ⟩ ∩ ⟨ A ∪ { b } ⟩ ⊆ ⟨ A ∪ { a ∧ b } ⟩ . Conversely if x ∈ ⟨ A ∪ { a ∧ b } ⟩ , then ( n ( a ∧ b ) ′ → x ′ ) ′ ∈ A for some ∈ N . Since a ∧ b ≤ a , b , then a ′ ≤ ( a ∧ b ) ′ and b ′ ≤ ( a ∧ b ) ′ . Using Theorem 2.12(10) repeatedly, we get n ( ( a ∧ b ) ′ ) → x ′ ≤ n ( a ′ ) → x ′ and n ( ( a ∧ b ) ′ ) → x ′ ≤ n ( b ′ ) → x ′ , which imply that ( n ( a ′ ) → x ′ ) ′ ≤ ( n ( ( a ∧ b ) ′ ) → x ′ ) ′ and ( n ( b ′ ) → x ′ ) ′ ≤ ( n ( ( a ∧ b ) ′ → x ′ ) ′ . Applying Theorem 3.19, we get ( n ( a ′ ) → x ′ ) ′ ∈ A and n ( b ′ ) → x ′ ) ′ ∈ A , i.e., x ∈ ⟨ A ∪ { a } ⟩ and x ∈ ⟨ A ∪ { b } ⟩ . Hence, x ∈ ⟨ A ∪ { a } ⟩ ∩ ⟨ A ∪ { b } ⟩ and so ⟨ A ∪ { a ∧ b } ⟩ ⊆ ⟨ A ∪ { a } ⟩ ∩ ⟨ A ∪ { b } ⟩ . □

Let A be an LI-ideal of IADL L. We define a binary relation “ ∼ ” on L as follows: x ∼ y if and only if ( x → y ) ′ ∈ A and ( y → x ) ′ ∈ A for all x , y ∈ L . Lemma 3.28

The binary relation “∼” on IADL L is an equivalent relation on L.

Proof.

Let L be an IADL L such that x , y , z ∈ L and A be an LI-ideal of L. Then 1.

x ∼ x ⇔ ( x → x ) ′ = m ′ = 0 ∈ A . This implies ∼ is reflexive on L.

x ∼ y ⇔ ( x → y ) ′ ∈ A and ( y → x ) ′ ∈ A if and only if ( y → x ) ′ ∈ A and ( x → y ) ′ ∈ A if and only if y ∼ x . Therefore ∼ is symmetric on L.

Assume x ∼ y and y ∼ z . We need to show that x ∼ z . Now x ∼ y and y ∼ z implies ( x → y ) ′ ∈ A and ( y → x ) ′ ∈ A , ( y → z ) ′ ∈ A and ( z → y ) ′ ∈ A . Since ( ( x → z ) ′ → ( x → y ) ′ ) ′ = ( ( x → y ) → ( x → z ) ) ′ ≤ ( y → z ) ′ it follows from Theorem 3.19 that ( ( x → y ) → ( x → z ) ) ′ ∈ A . Since ( x → y ) ′ ∈ A and A is LI-ideal of L, we have ( x → z ) ′ ∈ A .

Similarly, ( ( z → x ) ′ → ( z → y ) ′ ) ′ = ( ( z → y ) → ( z → x ) ) ′ ≤ ( y → x ) ′ it also follows from Theorem 3.19 that ( z → x ) ′ ∈ A . Therefore x ∼ z . Hence ∼ is transitive on L. □

Proposition 3.29

If x ∼ u and y ∼ v in an LI-ideal A of IADL L, then ( x → y ) ∼ ( u → v ) for any x , y , u , v ∈ L .

Proof.

Let L be an IADL and x , y , u , v ∈ L . Assume that x ∼ u and y ∼ v of an LI-ideal A. Then ( x → u ) ′ ∈ A and ( u → x ) ′ ∈ A , ( y → v ) ′ ∈ A and ( v → y ) ′ ∈ A . Since ( ( x → y ) → ( x → v ) ) ′ ≤ ( y → v ) ′ and ( ( x → v ) → ( x → y ) ) ′ ≤ ( v → y ) ′ , it follows from Theorem 3.19 that ( ( x → y ) → ( x → v ) ) ′ ∈ A and ( ( x → v ) → ( x → y ) ) ′ ∈ A . This implies that ( x → y ) ∼ ( x → v ) … ( 1 ) . Similarly Since ( ( x → v ) → ( u → v ) ) ′ ≤ ( u → x ) ′ ∈ A and ( ( u → v ) → ( x → v ) ) ′ ≤ ( x → u ) ′ ∈ A , it follows from Theorem 3.19 that ( ( x → v ) → ( u → v ) ) ′ ∈ A and ( ( u → v ) → ( x → v ) ) ′ ∈ A . This implies ( x → v ) ∼ ( ( u → v ) …(2). From (1), (2) and transitivity of ∼ , We conclude that ( x → y ) ∼ ( u → v ) . □

Corollary 3.30

Let a ∼ b and c ∼ d . Then 1.

a ′ ∼ b ′

( a ∨ c ) ∼ ( b ∨ d )

( a ∧ c ) ∼ ( b ∧ d )

Proof.

Let a , b , c , d ∈ L for an LI-ideal A of IADL L. Assume a ∼ b and c ∼ d . Then 1.

a ∼ b iff ( a → b ) ′ ∈ A and ( b → a ) ′ ∈ A iff ( b ′ → a ′ ) ′ ∈ A and ( a ′ → b ′ ) ′ ∈ A iff a ′ ∼ b ′ .

a ∼ b and c ∼ d implies ( a → c ) ∼ ( b → d ) from proposition 3.28. we also implies ( ( a → c ) → c ) ∼ ( ( b → d ) → d ) . Therefore a ∨ c ) ∼ ( b → d ) .

a ∼ b and c ∼ d implies ( a → c ) ∼ ( b → d ) . We also imply ( ( a → c ) → a ′ ) ∼ ( ( b → d ) → b ′ ) . Again we imply ( [ ( a → c ) → a ′ ] ′ ) ∼ ( [ b → d ) → b ′ ] ′ ) . Therefore ( a ∧ c ) ∼ ( b ∧ d ) . □

Theorem 3.31

The binary relation ∼ is a congruence relation on an IADL L.

Proof.

Using Lemma 3.28, Proposition 3.29 and Corollary 3.30, it is clear that ~ is a congruence relation on L. □

Now we have the following definition. Let A be an LI-ideal of IADL L. Then the equivalent class containing x is denoted by A _x such that Ax = { y ∈ L | x ∼ y } and the set of all equivalent classes of L is denoted by L | A such that L | A = { Ax | x ∈ L } . Lemma 3.32

A ₀ = A and A _m = { y ∈ L | y ′ ∈ A } .

Proof.

( y → 0 ) ′ = y ∈ A and ( 0 → y ) ′ ∈ A for all y ∈ L implies 0 ∼ y . This implies y ∈ A ₀. Therefore A ⊆ A ₀. Conversely, x ∈ A ₀ ⇒ ( 0 → x ) ′ ∈ A and ( x → 0 ) ′ ∈ A ⇒ 0 ∈ A and x ∈ A ⇒ x ∈ A . Therefore, A 0 ⊆ A . Hence, A ₀ = A.

A _m = { y ∈ L | m ∼ y } = { y ∈ L : ( m → y ) ′ ∈ A and ( y → m ) ′ ∈ A } = { y ∈ L : y ′ ∈ A and 0 ∈ A } = { y ∈ L : y ′ ∈ A } . □

Define binary operations “∪”, “∩”, “⇒” and unary operation “ N” on L / A as follows: A x ∪ A y = A x ∨ y A x ∩ A y = A x ∧ y , A x ⇒ A y = A x → y , A x N = A x ′ for all A x , A y ∈ L | A .

Then we have the following lemma. Lemma 3.33

Let L be an IADL and A be an LI-ideal of L. Then 1.

( L | A , ∪ , ∩ , A ₀ , A _m ) is a bounded ADL.

( L | A , ∪ , ∩ , ⇒ , N , A ₀ , A _m ) is an IADL which is called quotient implicative ADL.

Proof.

Let L be an IADL and A be an LI-ideal of L. Let A _x , A _y , A _z ∈ L | A for all x , y , z ∈ L . Then 1.

Since A ₀ ∪ A _x = A _{0∨
x} = A _x and A ₀ ∩ A _x = A _{0∧
x} = A ₀, for all x ∈ L . This implies A ₀ is the least element of L|A. Also A _m ∪ A _x = A _{m∨
x} = A _m and A _m ∩ A _x = A _{m∧
x} = A _x, for all x ∈ L . This implies A _m is the greatest element of L|A. Therefore, ( L | A , ∪ , ∩ , A ₀ , A _m ) is bounded. we can easily verify that ( L | A , ∪ , ∩ , A ₀ , A _m ) is an ADL.

To show ( L | A , ∪ , ∩ , ⇒ , N , A ₀ , A _m ) is an IADL,

(a)

A _x ∪ A _y = A _x∨y = A _(x→y)→y = ( A _x ⇒ A _y ) ⇒ A _y

(b)

A x ∩ A y = A x ∧ y = [ A ( x → y ) → x ′ ] = A _(x→y)→x′ = ( Ax ⇒ Ay ) ⇒ Ax ′ = [( A _x ⇒ A _y ) ⇒ A ^N ] _x ^N

(c)

A _x ⇒ ( A _y ⇒ A _z ) = A _x→(y→z) = A _y→(x→z) = A _y ⇒ ( A _x ⇒ A _z )

(d)

A _m ⇒ A _x = A _m→x = A _x

(e)

A _x ⇒ A _m = A _x→m = A _m

(f )

A _x ⇒ A _y = A _x→y = A _y′ _→x′ = A _y′ ⇒ A _x′ = A y ^N ⇒ A x ^N

(g)

A 0 N = A ₀ ′ = A _m

Therefore, ( L | A , ∪ , ∩ , ⇒ , N , A ₀ , A _m ) is quotient IADL. □

Conclusion

Many scholars investigated ideals in lattice and ADL. Furthermore, LI-ideals in Lattice implication algebras are discussed by different researchers. For the advancement of many valued logical algebra we extend this notion to ADL concept. That is, first we investigated H-implicative almost distributive lattice. Then we discussed LI-ideals in IADL. Lastly, we found an interesting result that set theoretic LI-ideals is an LI-ideal and some characterization of LI-ideals in IADL. In the future we will have fuzzy version of this notion.

Author contributions

All the authors are contributed equally in this manuscript and also both authors read and approved the final manuscript.

Ethics and consent

Ethics and consent were not required.

Data availability

No data are associated with this article.

References 1

Assaye

Alamneh

Mekonnen

: Implicative Almost Distributive Lattice. Int. J. Comput. Sci. Appl. Math. 2018;4(1):19–22.

Assaye

Alamneh

Mekonnen

: Positive implicative and associative filters of implicative almost distributive lattice. Bull. Inter. Math. Virtual Inst. 2019;9(1):111–119.

Jun

: Implicative filters of lattice implication algebras. Bull. Korean Math. Soc. 1997;34(2):193–198.

Jun

Roh

: LI-ideals in lattice implication algebras. Bull. Korean Math. Soc. 1998;35(1):13–24.

Kolluru

Bekele

: Implicative algebras. Momona Ethiop. J. Sci. 2012;4(1):90–101. 10.4314/mejs.v4i1.74059

Swamy

Rao

: Almost Distributive Lattices. J. Aust. Math. Soc. (Series A). 1981;31(1):77–91. 10.1017/S1446788700018498

Qin

: Lattice H-implication algebras and lattice implication algebra classes. J. Hebin Mining Civil Eng. Institute. 1992;2:139–143.

: Lattice implication algebras. J. South West Jiaotong University. 1993;28(1):20–27.

Ruan

Qin

: Lattice-Valued Logic: An Alternative Approach to Treat Fuzziness and Incomparability. Berlin: Springer;2003.

10.5256/f1000research.174866.r421472

Reviewer response for version 1

Pelaitay

Gustavo

1 Referee https://orcid.org/0000-0002-5935-6808 1Universidad Nacional de San Juan, San Juan, Argentina

Competing interests: No competing interests were disclosed.

14 10 2025

2025

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

approve-with-reservations

COMPREHENSIVE REVIEW REPORT

The article "Li-ideals of implicative almost distributive lattices" presents a thoughtful and valuable investigation into lattice ideals within implicative almost distributive lattices. The authors have successfully extended the concept of LI-ideals from lattice implication algebras to IADLs, introducing H-implicative almost distributive lattices and establishing important relationships between different algebraic structures. The work demonstrates strong mathematical reasoning and makes a genuine contribution to the field.

DETAILED RESPONSES TO REVIEW QUESTIONS:

1. CLARITY OF PRESENTATION AND LITERATURE:

The mathematical content is well-structured and technically sound. To make the work even more accessible, we suggest enhancing the transitions between sections and adding brief explanatory notes to help readers follow the technical developments. The reference list could be strengthened by including some recent work on ideals in residuated lattices, which would provide additional context for your contributions.

2. STUDY DESIGN AND TECHNICAL SOUNDNESS:

The research approach is appropriate and the proofs are rigorous. The logical progression from definitions to main results follows good mathematical practice, and the examples effectively illustrate the concepts.

3. METHODS AND REPRODUCIBILITY:

The paper contains all necessary mathematical elements. To help other researchers build upon this work, consider adding a brief overview in the introduction that outlines how the main results connect. Breaking down some of the more complex proofs into intermediate steps and including additional examples of key constructions would make the work even more approachable.

4. STATISTICAL ANALYSIS:

Not applicable - this is appropriately a theoretical mathematical work.

5. SOURCE DATA AVAILABILITY:

All necessary information is contained within the paper.

6. CONCLUSIONS:

The conclusions are well-supported by the results and accurately reflect the research findings.

SUGGESTIONS FOR ENHANCEMENT:

We believe these adjustments would further strengthen your already solid work:

- Expand the literature review to include connections to contemporary work in related algebraic structures

- Add transitional explanations to guide readers through the technical developments

- Consider providing additional examples that illustrate the practical implications of your theoretical results

- Ensure consistent notation throughout the document

OVERALL RECOMMENDATION: ACCEPT AFTER MINOR REVISIONS

This paper makes a valuable contribution to lattice theory and algebraic logic. The mathematical foundation is strong and the results are meaningful. With attention to the suggested enhancements, this work will be an excellent addition to the literature and will be well-received by the research community.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Reviewer Expertise:

Algebraic Logic

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

10.5256/f1000research.174866.r368840

Reviewer response for version 1

Yang

1 Referee Yi

Liu

2 Co-referee 1Southwest Jiaotong University, Chengdu, China 2College of Mathematics and Information Sciences, Neijiang Normal University, Neijiang, Sichuan, China

Competing interests: No competing interests were disclosed.

21 3 2025

2025

recommendation

approve

Ideals and Filters of two basic research direction in logical algebra. In current paper, authors investigated the LI-ideals of ADL concepts. For the advancement of many valued logical algebra we extend this notion to ADL concept. Firstly, authors investigated H-implicative almost distributive lattice; LI-ideals in IADL are discussed. Authors also found

that set theoretic LI-ideals is an LI-ideal and some characterizations of LI-ideals in IADL.

All results are soundness. The proofs are correct. So I think this paper can be accepted subjected to a minor revised. Some references regarding the ideals and filters in residued lattices should be added.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Reviewer Expertise:

logical algebras

We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.