EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5532 ISSN 1307-5543 – ejpam.com Published by New York Business Global L-Convex Sublattice of an L-Lattice and Complement of an L-set Aparna Jain1, Iffat Jahan2 ∗ 1 Department of Mathematics, Shivaji College, University of Delhi, Delhi, India 2 Department of Mathematics, Ramjas College, University of Delhi, Delhi, India Abstract. In this paper, the authors define and explore the notion of an L-convex sublattice in an L-lattice. The investigations in this paper lead to ’The Unique Representation Theorem’ for L-convex sublattices. Also, the authors effectively use the concept of order reversing involution on the lattice L of truth values to define complement of an L-set. Further, they employ this notion in the studies of L- prime ideals and L-maximal ideals. 2020 Mathematics Subject Classifications: 06B10, 06D72, 06D75, 08A72 Key Words and Phrases: Lattices, generated L-sublattice, generated L-ideal, generated L-dual ideal, L-convex sublattice, Complement of an L-set, L-maximal ideal, L-prime ideal 1. Introduction The literature on fuzzy algebraic structures has been growing ever since the introduction of the concept of a fuzzy subgroup by A. Rosenfeld [14] in the year 1971. Ajmal and Thomas [4–6] systematically developed the theory of fuzzy sublattices in a lattice. They introduced the notions of a fuzzy sublattice, fuzzy ideal (dual ideal), fuzzy prime ideal (dual ideal), fuzzy ideal (dual ideal) generated by a fuzzy set and studied their properties. The concept of a fuzzy convex sublattice was also introduced by Ajmal and Thomas in [4, 5], wherein the Unique Representation Theorem for convex sublattices was extended to fuzzy setting. The concept of an L-fuzzy set was pioneered by Goguen [7] in the year 1967 .In [10], the authors studied the concept of an L-lattice. This shifts their studies from the evalua- tion lattice [0, 1] to a more general lattice L. Moreover in [10], authors made one more transition by studying the notions of L-substructures in an L-lattice instead of fuzzy sub- structures of an ordinary lattice. Thus, the parent structure also shifts from a lattice to an L-lattice. It is worthwhile to mention here that under this arrangement, some notions ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5532 Email addresses: jainaparna@yahoo.com (A. Jain), ij.umar@yahoo.com (I. Jahan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 2 of 20 such as L-maximal ideal can be defined more meaningfully in the L-setting. In past few years, Ajmal and Jahan have successfully developed the theory of L-subgroups in [1–3, 8, 9, 12]]. They have taken the theory of L-subgroups towards completion by studying the concepts of characteristic subgroups, normalizer of a subgroup, nilpotent subgroups, solvable subgroups, normal closure of a subgroup etc., within the framework of L-groups. In [9], Jahan and Manas studied maximal and Frattini L-subgroups of an L-group. In [10], the notions of an L-maximal ideal and L-prime ideal in an L-lattice are defined and various related results are studied. In order to take such studies further, in the present paper, we introduce the notion of an L-convex sublattice in an L-lattice.Then this notion of convex L-sublattice is used to demonstrate that the Unique Representation Theorem of classical lattice theory for convex sublattices also holds under the L-setting wherein the parent structure is an L-lattice. In the last section of this paper, we use the notion of an order reversing involution on a lattice to define the concept of complement of an L-set. The notion of order reversing involution occurs frequently in fuzzy topological spaces and fuzzy implication algebras [11, 13, 15–17]. Thereafter, we establish some significant analogues of results of classical lattice theory to L-setting using complement of an L-set, thereby taking the theory of L-lattices to a more developed stage. 2. Preliminaries In this work, (M,≤,∧,∨) denotes a bounded lattice and (L,≤,∧,∨) a complete lattice. The maximal and minimal elements of both the lattices L and M are denoted by 1 and 0 respectively. The notations ’≤’, ’∧’ and ’∨’ denote the partial order, meet and join operations respectively of both the lattices L and M . An L-subset of M is defined as a mapping µ : M → L. The collection of all L-subsets of M is denoted by LM and is called the L-power set of M . If µ, η ∈ LM , η is said to be contained in µ(denoted by η ⊆ µ), if η(x) ≤ µ(x), ∀ x ∈ M . Moreover, η is said to be properly contained in µ, if η ⊆ µ and there exists x ∈ M such that η(x) < µ(x). If η ⊆ µ, then η is said to be an L-subset of an L-set µ. The set of all L-subsets of µ is called the L-power set of µ and is denoted by Lµ. If µ ∈ LM and α ∈ L, the level subset µα and the strong level subset µ> α are defined as follows: µα = {x ∈ M/µ(x) ≥ α} and µ> α = {x ∈ M/µ(x) > α}. Clearly, µ> α ⊆ µα, ∀ α ∈ L and if α ≤ β in L, then µβ ⊆ µα and µ> β ⊆ µ> α . A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 3 of 20 If µ ∈ LM , then ∨x∈Mµ(x) and ∧x∈Mµ(x) are called the tip and tail of µ, respectively. The arbitrary union ∪i∈I(µi) and intersection ∩i∈I(µi) of a family {µi}i∈I of L-subsets of M are given by: (∪i∈Iµi)(x) = ∨i∈Iµi(x) and (∩i∈Iµi)(x) = ∧i∈Iµi(x). Definition 1 ([4]). Let µ ∈ LM . Then, µ is said to be an L-sublattice of M if ∀ x, y ∈ M µ(x ∨ y) ≥ µ(x) ∧ µ(y) and µ(x ∧ y) ≥ µ(x) ∧ µ(y). Let L(M) denote the set of all L-sublattices of M . If µ ∈ L(M), µ is called an L-Lattice and is denoted by L(µ,M). If µ, η ∈ L(M) and η ⊆ µ, then η is called an L-sublattice of the L-lattice µ. The collection of all L-sublattices of µ is denoted by L(µ). In this paper, we shall study the L-convex sublattices of an L-lattice µ rather convex sublattices of an ordinary lattice. The following theorems provide the level subset characterizations and strong level subset characterizations of an L-sublattice of µ. For similar characterizations of L-sublattices of M , we refer to [10]. Theorem 1 ([10]). Let µ, η ∈ LM be such that η ⊆ µ. Also, let L(µ,M) be an L-lattice and ao = tip{η}. Then, η is an L-sublattice of µ if and only if each level subset ηα is a sublattice of µα, ∀α ≤ ao. Equivalently, η is an L-sublattice of µ if and only if each nonempty level subset ηα is a sublattice of µα. Theorem 2 ([10]). Let L be a chain. Let µ, η ∈ LM be such that η ⊆ µ. Also, let L(µ,M) be an L-lattice and ao = tip{η}. Then, η is an L-sublattice of µ if and only if each strong level subset η>α is a sublattice of µ> α , ∀ α < ao. Equivalently, η is an L-sublattice of µ if and only if each nonempty strong level subset η>α is a sublattice of µ> α . The notions of L-ideal, L-dual ideal in lattice M and L-ideal, L-dual ideal in an L-lattice µ are defined as follows: Definition 2 ([10]). Let µ ∈ LM . Then, [(i)]µ is called an L-ideal ofM if µ ∈ L(M) and x ≤ y inM implies µ(x) ≥ µ(y) in L; µ is called an L-dual ideal of M if µ ∈ L(M) and x ≤ y in M implies µ(x) ≤ µ(y) in L. Definition 3 ([10]). Let µ, η ∈ LM be such that η ⊆ µ. Also, let L(µ,M) be an L-lattice. Then, [(i)]η is called an L-ideal of µ if η(x ∨ y) ≥ η(x) ∧ η(y) and η(x ∧ y) ≥ µ(x) ∧ η(y); ∀x, y ∈ M. η is called an L-dual ideal of µ if η(x ∧ y) ≥ η(x) ∧ η(y) and η(x ∨ y) ≥ η(x) ∧ µ(y); ∀x, y ∈ M. A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 4 of 20 It is important to note here that in a bounded lattice M , an L-ideal attains its supre- mum at the least element of M , whereas an L-dual ideal attains its supremum at the greatest element of M . The following theorems provide the level subset characterizations and strong level subset characterizations of an L-ideal (L-dual ideal) of µ. For similar characterizations of L-ideal (L-dual ideal) of M , refer to [10]. Theorem 3 ([10]). Let µ, η ∈ LM be such that η ⊆ µ. Also, let L(µ,M) be an L-lattice and ao = tip{η}. Then, η is an L-ideal(L-dual ideal) of µ if and only if each level subset ηα is an ideal(dual ideal) of µα, ∀α ≤ ao. Equivalently, η is an L-ideal(L-dual ideal) of µ if and only if each nonempty level subset ηα is an ideal(dual ideal) of µα. Theorem 4 ([10]). Let L is a chain. Let µ, η ∈ LM be such that η ⊆ µ. Also, let L(µ,M) be an L-lattice and ao = tip{η}. Then, η is an L-ideal(L-dual ideal) of µ if and only if each strong level subset η>α ∀α < ao, is an ideal (dual ideal) of µ> α . Equivalently, η is an L-ideal(L-dual ideal) of µ if and only if each nonempty strong level subset η>α is an ideal (dual ideal) of µ> α . It can be easily verified that the intersection of an arbitrary family of L-sublattices(L- ideals, L-dual ideals) of µ is an L-sublattice(L-ideal, L-dual ideal) of µ. This leads to the definition of an L-sublattice(L-ideal, L-dual ideal) generated by an L-subset η of µ as the intersection of all L-sublattices(resp. L-ideals, L-dual ideals) of µ containing η. These are denoted by [η]µ, (η]µ and [η)µ respectively. The following result from [3] gives the struc- tural compositions of an L-sublattice, L-ideal, L-dual ideal of an L-lattice µ generated by an L-subset η of µ in terms of level subsets. Theorem 5 ([10]). Let L(µ,M) be an L-lattice, η ∈ LM , η ⊆ µ with ao = tip{η}. [(i)]Define an L-subset ηo of M as: ηo(x) = ∨ t≤ao {t : x ∈ [ηt]}, where [ηt] is a sublattice of µt generated by ηt. Then, ηo is an L-sublattice of µ and ηo = [η]µ. Define an L-subset η1 of M as: η1(x) = ∨ t≤ao {t : x ∈ (ηt]}, where (ηt] is an ideal of µt generated by ηt. Then, η1 is an L-ideal of µ and η1 = (η]µ. Define an L-subset η2 of M as: η2(x) = ∨ t≤ao {t : x ∈ [ηt)}, where [ηt) is an dual ideal of µt generated by ηt. Then, η2 is an L-dual ideal of µ and η2 = [η)µ. A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 5 of 20 The concept of a maximal ideal could be meaningfully extended from classical setting to fuzzy setting by the authors in [10] by shifting the parent structure from classical lattice to an L-structure as follows: Definition 4 ([10]). Let L(µ,M) be an L-lattice. [(i)]A proper L-ideal η of µ is called an L-maximal ideal of µ if for any L-ideal θ of µ, whenever η ⊆ θ ⊆ µ, then θ = η or θ = µ. A proper L-dual ideal η of µ is called an L-dual maximal ideal of µ if for any L-dual ideal θ of µ, whenever η ⊆ θ ⊆ µ, then θ = η or θ = µ. In [10], some characterizations of an L-maximal ideal and L-maximal dual ideal of µ were provided. Further, an L-prime ideal(L-prime dual ideal) in lattice M and an L-prime ideal(L-prime dual ideal) in an L-lattice µ are defined as follows: Definition 5 ([4]). [(i)] (i) An L-ideal µ of M is called an L-prime ideal of M if µ(x ∧ y) ≤ µ(x) ∨ µ(y); ∀x, y ∈ M. (ii) An L-dual ideal µ of M is called an L-prime dual ideal of M if µ(x ∨ y) ≤ µ(x) ∨ µ(y); ∀x, y ∈ M. Definition 6 ([10]). Let L(µ,M) be an L-lattice. [(i)]A proper L-ideal η of µ is called an L-prime ideal of µ if, ∀ x, y ∈ M η(x ∧ y) ∧ µ(x) ∧ µ(y) ≤ η(x) or η(x ∧ y) ∧ µ(x) ∧ µ(y) ≤ η(y). A proper L-dual ideal η of µ is called an L-dual prime ideal of µ if, ∀ x, y ∈ M η(x ∨ y) ∧ µ(x) ∧ µ(y) ≤ η(x) or η(x ∨ y) ∧ µ(x) ∧ µ(y) ≤ η(y). In [10], the authors defined a fuzzy convex sublattice of a lattice and studied the related properties. On similar lines, an L-convex sublattice of a lattice M can be defined as follows: Definition 7. If µ ∈ L(M), then µ is called an L-convex sublattice of M if for each interval [a, b] ⊆ M , µ(x) ≥ µ(a) ∧ µ(b), ∀ x ∈ [a, b]. A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 6 of 20 3. L-convex Sublattice of an L-lattice In this section, L is taken to be a complete and completely distributive lattice in some results. The definition of a completely distributive lattice is well known in the literature and can be found in any standard text on the subject. Let {Ji : i ∈ I} be any family of subsets of a complete lattice L and F denote the set of choice functions for Ji, i.e. functions f : I → ∏ i∈I Ji such that f(i) ∈ Ji for each i ∈ I. Then, we say that L is a completely distributive lattice, if ∧{∨ i∈I Ji } = ∨ f∈F {∧ i∈I f(i) } . The above law is known as the complete distributive law. Thus, in order theory, a complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets. Note that the dual of completely distributive law is valid in a completely distributive lattice. We begin this section by defining an L-convex sublattice of an L-lattice µ and study its properties. Definition 8. Let L(µ,M) be an L-lattice. An L-sublattice η of µ is called an L-convex sublattice of µ if η(x) ≥ η(a) ∧ η(b) ∧ µ(x) where a ≤ x ≤ b in M. The following characterisations of an L-convex sublattice η of µ with the help of level subsets and strong level subsets of η can be verified easily. Theorem 6. Let L(µ,M) be an L-lattice and η ∈ L(µ) with ao = tip{η}. Then, η is an L-convex sublattice of µ if and only if each level subset ηt, ∀ t ≤ ao, is a convex sublattice of µt. Equivalently, η is an L-convex sublattice of µ if and only if each nonempty level subset ηt is a convex sublattice of µt. Theorem 7. Let L be a chain. Let L(µ,M) be an L-lattice and η ∈ L(µ) with ao = tip{η}. Then, η is an L-convex sublattice of µ if and only if each strong level subset η>t is a convex sublattice of µ> t , ∀t < ao. Equivalently, η is an L-convex sublattice of µ if and only if each nonempty strong level subset η>t is a convex sublattice of µ> t . We now provide the following examples of L-convex sublattices in an L-lattice: Example 1. Let M = ℵ be the chain of natural numbers and L = P (ℵ), the power set of ℵ, be the Boolean Algebra. Define the following L-subsets of ℵ : η(n) = { ∅ if n = 1, {1, 2, ..., n− 1} ∀n ≥ 2; A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 7 of 20 and µ(n) = {1, 2, ..., n} ∀n ∈ ℵ. Then η ⊆ µ. Moreover, it is easy to verify that η and µ are L-sublattices of ℵ. Futhermore, η turns out to be an L-convex sublattice of µ. It is wothwhile to note here that the set of all level subsets of η form a chain in the lattice M = ℵ. Here, we write Ai = {1, 2, · · · , i− 1} ∀i ≥ 2. Further, note that ηAi = Ai and η∅ = ∅. Thus, we have A1 = ∅ ⊆ A2 ⊆ A3 ⊆ · · · ⊆ An ⊆ · · · ⊆ ℵ. Example 2. Let X = {a, b, c} and L = P (X) be the power set of X. Then ⟨L, ∩, ∪, ′⟩ is a Boolean Algebra where ′∪′, ′∩′ and ′′′ denote the ordinary intersection, union and complement of members of L respectively. Further, It is easy to see that L is Boolean Algebra with order reversing involution given by : τ : L −→ L∗, τ(A) = A′. Let M = {1, 2, 3, 6} denote the set of all factors of ′6′. Then ⟨M,∨,∧, ′⟩, where a ∨ b = lcm{a, b}, a ∧ b = gcd{a, b} and a′ = 6 a ; ∀ a, b ∈ M , is also a Boolean Algebra. In the following diagram, (i) and (ii) represent Boolean Algebras M and L respectively. (i) (ii) Define the following L-subsets µ and η of M : µ(A) = { 2 if A ∈ {∅, {a}, {b}, {a, b}}, 6 if A = P (X) \ {∅, {a}, {b}, {a, b}}; A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 8 of 20 and η(A) =  1 if A ∈ {∅, {b}}, 2 if A ∈ {{a}, {a, b}}, 3 if A ∈ {c, {{b, c}}, 6 if A ∈ {{a, c}, X}. Now, note that η ⊆ µ. The set {ηa : a ∈ Im η} of all level subset of η is determined below : η1 = M, η2 = {{a}, {a, b}, {a, c}, X}, η3 = {{c}, {b, c}, {a, c}, X} and η6 = {{a, c}, X}. Further, the set {µa : a ∈ Im µ} of all level subset of µ is determined below : µ2 = M, and µ6 = {{c}, {b, c}{a, c}, X}. Now, it is easy to see that η and µ are L-sublattices of M . Futhermore, η forms an L- convex sublattice of µ. Observe that in this example the set of all level subsets of η does not form a chain. Infact, the set of all level subsets {ηa : a ∈ Im η} turns out to be only a poset under the ordering of usual set theoretic containment. Example 3. Let M = ∅ ∪ Z ∪ {{n} : n ∈ Z}. Then M is a Boolean Algebra with the following Hasse Diagram : Further, let L = {A ⊆ R : either A or A′ is finite}. Here A′ is complement of A in R. It is easy to see that L is Boolean Algebra with order reversing involution given by : τ : L −→ L∗, τ(A) = A′. Define the following L-subsets of M : η(A) =  ∅ if A = Z, R if A = ∅, {n} if n ∈ Z; A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 9 of 20 and µ(A) =  ∅ if A = Z, R if A = ∅, {n,−n} if n ∈ Z. Now, note that η ⊆ µ. The set {ηa : a ∈ Im η} of all level subset of η is determined below : ηR = ∅, η{n} = {Z, {n}} and η∅ = M. Further, the set {µa : a ∈ Im µ} of all level subset of µ is determined below : µR = ∅, µ{± n} = {Z, {n}}, and µ∅ = M. Now, it is easy to see that η and µ are L-sublattices of M . Futhermore, η forms an L- convex sublattice of µ. Observe that in this example the Hesse Diagram of set of all level subsets of both η and µ coincide with that of Hasse Diagram of the lattice M given above. Infact, the set of all level subsets {ηa : a ∈ Im η} turns out to be lattice under the usual set theoretic containment. The following result is also straightforward. Theorem 8. The intersection of an arbitrary family of L-convex sublattices of L-lattice µ is an L-convex sublattice of µ. The above result is instrumental in defining an L-convex sublattice of µ generated by an L-subset η of µ. Definition 9. An L-convex sublattice of L-lattice µ generated by an L-subset η of µ is defined as the intersection of all L-convex sublattices of µ containing η and is denoted by [η]cµ. Thus, [η]cµ = ⋂ {ηi : ηi is an L-convex sublattice of µ, η ⊆ ηi, ∀ i ∈ I}. The next result provides a complete structure of L-convex sublattice generated by L-subset η of µ in terms of level subsets. Theorem 9. Let L be a complete and completely distributive lattice and L(µ,M) be an L-lattice. Let η ∈ LM with η ⊆ µ and a0 = tip{η}. Define an L-subset η′ of M as: η′(x) = ∨ t≤ao {t : x ∈ [ηt]c}, ∀ x ∈ M ; where [ηt]c is the convex sublattice of lattice µt generated by ηt. Then, η′ is an L-convex sublattice of µ and η′ = [η]cµ. A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 10 of 20 Proof. Since η ⊆ µ, ηt ⊆ µt, ∀ t ∈ L. As µ is an L-lattice, µt is a sublattice of M , ∀ t ≤ tip{µ}. Moreover, [ηt]c ⊆ µt, as [ηt]c is the convex sublattice of µt generated by ηt. Thus, η′(x) = ∨ t≤ao {t : x ∈ [ηt]c} ≤ ∨ t≤tip{µ} {t : x ∈ µt} = µ(x). We thus have η′ ⊆ µ. Further, to prove that η ⊆ η′, let x ∈ M and let η(x) = α ≤ a0. Then, x ∈ ηα ⊆ [ηα]c. Therefore, by definition of η′, α ≤ η′(x). That is, η(x) ≤ η′(x). Thus, η ⊆ η′. We now prove that η′ is an L-sublattice of µ. For any z ∈ M , define a subset Lη(z) of L as follows: Lη(z) = {t ∈ L/t ≤ a0, z ∈ [ηt]c}. Clearly, η′(x) = ∨ Lη(x). Let x, y ∈ M , a ∈ Lη(x) and b ∈ Lη(y). We claim that a ∧ b ∈ Lη(x ∨ y). First note that ηa ∪ ηb ⊆ ηa∧b. Since a ∈ Lη(x) and b ∈ Lη(y), we have a, b ≤ a0, x ∈ [ηa]c, y ∈ [ηb]c. Therefore, x = p{xi} (a lattice polynomial in variables xi’s. where xi ∈ ηa, ∀i). Similarly, y = q{yj} (a lattice polynomial in variables yj ’s. where yj ∈ ηb, ∀j). Thus, x ∨ y is also a lattice polynomial in variables xi’s and yj ’s, where xi, yj ∈ ηa ∪ ηb ⊆ ηa∧b. That is, x ∨ y ∈ [ηa∧b]c. We also have a ∧ b ≤ a0. Thus, a ∧ b ∈ Lη(x ∨ y). This implies that η′(x ∨ y) ≥ a ∧ b; ∀ a ∈ Lη(x) and b ∈ Lη(y). Consequently, η′(x ∨ y) ≥ ∨{a ∧ b/a ∈ Lη(x), b ∈ Lη(y)} = {∨{a/a ∈ Lη(x)}} ∧ {∨{b/b ∈ Lη(y)}} (as L is a completely distributive lattice) = η′(x) ∧ η′(y). Similarly, it can be proved that η′(x ∧ y) ≥ η′(x) ∧ η′(y); ∀x, y ∈ M. Thus, η′ is an L-sublattice of µ. Further, to establish that η′ is an L-convex sublattice of µ, we shall prove that, (i)(ii) η′(w) ≥ η′(x) ∧ η′(y) ∧ µ(w) where x ≤ w ≤ y in M . A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 11 of 20 For any z ∈ M , let Lη(z) = {t ∈ L/t ≤ a0, z ∈ [ηt]c}. Then, η′(x) = ∨ Lη(x). Also let Lµ(z) = {t ∈ L/ t ≤ tip {µ}, z ∈ µt = [µt]}. Let r ∈ Lµ(w), s ∈ Lη(x) and t ∈ Lη(y). Then, r ≤ tip {µ}; s, t ≤ a0, w ∈ [µr] = µr, x ∈ [ηs]c and y ∈ [ηt]c. Hence, s ∧ t ∧ r ≤ a0 and we have x ∈ [ηs]c ⊆ µs ⊆ µs∧t∧r; y ∈ [ηt]c ⊆ µt ⊆ µs∧t∧r; and w ∈ [µr] = µr ⊆ µs∧t∧r. Thus, x, y, w ∈ µs∧t∧r. Moreover, x ∈ [ηs]c ⊆ [ηs∧t∧r]c and y ∈ [ηt]c ⊆ [ηs∧t∧r]c; and [ηs∧t∧r]c is a convex sublattice of µs∧t∧r generated by the L-set ηs∧t∧r. Therefore, we have w ∈ [ηs∧t∧r]c (as x ≤ w ≤ y in M). This implies η′(w) ≥ s ∧ t ∧ r; ∀ r ∈ Lµ(w), s ∈ Lη(x) and t ∈ Lη(y). That is, η′(w) ≥ ∨{s ∧ t ∧ r/r ∈ Lµ(w), s ∈ Lη(x) and t ∈ Lη(y)} = {∨{s/s ∈ Lη(x)}} ∧ {∨{t/t ∈ Lη(y)}} ∧ {∨{r/r ∈ Lµ(w)}} (as L is a completely distributive lattice) = η′(x) ∧ η′(y) ∧ µ(w). Hence, η′ is an L-convex sublattice of µ. Now it is left to prove that η′ is the smallest L-convex sublattice of µ containing η. For this, suppose θ is an L-convex sublattice of µ such that η ⊆ θ. Then, ηt ⊆ θt, ∀ t ∈ L. Since θ is an L-convex sublattice of µ, therefore by Theorem 6, each nonempty θt is a convex sublattice of µt. Therefore, [θt]c = θt. This implies that [ηt]c ⊆ θt, ∀ t ≤ a0. Also, a0 = tip{η} ≤ tip{θ}. Thus, ∀ x ∈ M , we have η′(x) = ∨ t≤ao {t : x ∈ [ηt]c} ≤ ∨ t≤tip{θ} {t : x ∈ θt} A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 12 of 20 = θ(x). That is, η′ ⊆ θ. Hence, η′ = [η]cµ. The next result is significant for establishing the Unique Representation Theorem for L-convex sublattice of an L-lattice µ. The proof being trivial is omitted. Theorem 10. Let L(µ,M) be an L-lattice and η ⊆ µ be an L-ideal (L-dual ideal) of µ. Then, η is an L-convex sublattice of µ. Before discussing the Unique Representation Theorem for L-convex sublattices in an L- lattice µ, we provide the structural composition of an L-ideal of µ generated by an L-subset η of µ in terms of strong level subsets of η. The following result is proved by taking L to be a dense chain. Note that a dense chain is a completely distributive lattice. Theorem 11. Let L be a dense chain and L(µ,M) be an L-lattice. Let η ∈ LM , η ⊆ µ and a0 = tip{η}. Define an L-set η̂ of M as follows: η̂(x) = ∨tt ]}, where (η>t ] is an ideal of µ> t generated by η>t . Then, η̂ = (η]µ. Proof. Since η ⊆ µ, η>t ⊆ µ> t , ∀ t ∈ L. As µ is an L-lattice, µ> t is a sublattice of M , ∀ t < tip{µ}. Also, as (η>t ] is an ideal of µ> t generated by η>t , we have (η>t ] ⊆ µ> t , ∀ t < a0. Thus, η̂(x) = ∨tt ]} ≤ ∨t t } ≤ µ(x). We thus have η̂ ⊆ µ. To establish that η ⊆ η̂, we prove that η>α ⊆ (η̂)>α , ∀ α ∈ L. Let α ∈ L and x ∈ η>α . Then, η(x) > α. Since L is a dense chain, ∃ β ∈ L such that η(x) > β > α. This implies x ∈ η>β and hence x ∈ (η>β ]. Consequently, η̂(x) = ∨tt ]} ≥ β > α. That is, x ∈ (η̂)>α . This proves that η ⊆ η̂. We now prove that η̂ is an L-ideal of µ. For any z ∈ M , define a set Lη(z) = {t ∈ L/t < a0, z ∈ (η>t ]}. Then, η̂(x) = ∨ Lη(x). Let x, y ∈ M . We claim that for any a ∈ Lη(x) and b ∈ Lη(y), a ∧ b ∈ Lη(x ∨ y). Suppose, a ∈ Lη(x) and b ∈ Lη(y). Then, a < a0, b < b0, x ∈ (η>a ], y ∈ (η>b ]. A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 13 of 20 Since L is a chain, a ∧ b < a0. Now, x ∈ (η>a ], y ∈ (η>b ] implies that x ≤ x1 ∨ . . . ∨ xn, xi ∈ η>a , ∀ i; and y ≤ y1 ∨ . . . ∨ ym, yj ∈ η>b , ∀ j. Then, we have x ∨ y ≤ (∨xi) ∨ (∨yj), which is a finite join of elements of η>a ∪ η>b and η>a ∪ η>b ⊆ η>a∧b. Therefore, x ∨ y ∈ (η>a∧b] and a ∧ b < a0. Thus, a ∧ b ∈ Lη(x ∨ y). Consequently, η̂(x ∨ y) ≥ a ∧ b; ∀ a ∈ Lη(x) and b ∈ Lη(y). Hence, η̂(x ∨ y) ≥ ∨{a ∧ b/a ∈ Lη(x), b ∈ Lη(y)} = {∨{a/a ∈ Lη(x)}} ∧ {∨{b/b ∈ Lη(y)}} (as L is a completely distributive lattice) = η′(x) ∧ η′(y). Now, to verify that η̂(x ∧ y) ≥ µ(x) ∧ η̂(y), we again define the following subsets of L for z ∈ M : Lη(z) = {t ∈ L/t < a0, z ∈ (η>t ]} and Lµ(z) = {t ∈ L/t ≤ tip{µ}, z ∈ µt = [µt]}. Thus, η̂(x) = ∨ Lη(x) and µ(x) = ∨ Lµ(x). If a ∈ Lµ(x) and b ∈ Lη(y), then a ≤ tip {µ}, b < a0, x ∈ µa = [µa] and y ∈ (η>b ]. Therefore, a ∧ b < a0 and y ≤ y1 ∨ . . . ∨ ym, yj ∈ η>b , ∀ j. This implies x ∧ y ≤ y1 ∨ . . . ∨ ym, yj ∈ η>b ⊆ η>a∧b, ∀ j. We thus have x ∧ y ∈ (η>b ] ⊆ (η>a∧b] and therefore, a ∧ b ∈ Lη(x ∧ y); ∀ a ∈ Lµ(x) and b ∈ Lη(y). That is, η̂(x ∧ y) ≥ a ∧ b; ∀ a ∈ Lµ(x) and b ∈ Lη(y). Consequently, η̂(x ∧ y) ≥ ∨{a ∧ b/a ∈ Lµ(x) and b ∈ Lη(y)} = {∨{a/a ∈ Lµ(x)}} ∧ {∨{b/b ∈ Lη(y)}} (as L is a completely distributive lattice) = µ(x) ∧ η̂(y). A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 14 of 20 We thus get that η̂ is an L-ideal of µ. Finally, to prove that η̂ is the smallest L-ideal of µ containing η, suppose θ is an L-ideal of µ such that η ⊆ θ. Then, η>t ⊆ θ>t and θ>t is an ideal of µ> t . Therefore, (θ > t ] = θ>t . We thus have: η̂(x) = ∨tt ]} ≤ ∨ t≤tip{θ} {t : x ∈ θ>t } ≤ θ(x). That is, η̂ ⊆ θ. Hence, η̂ = (η]µ. A similar result holds for L-dual ideal generated by η. Theorem 12. Let L be a dense chain and L(µ,M) be an L-lattice. Let η ∈ LM , η ⊆ µ and a0 = tip{η}. Define an L-subset η̌ of M as follows: η̌(x) = ∨tt )}, where [η>t ) is a dual ideal of µ> t generated by η>t . Then, η̌ = [η)µ. We now establish the Unique Representation Theorem for L-convex sublattices in an L- lattice. In the following theorem, O represents the constant L-subset with all truth values equal to 0 of lattice L. Theorem 13. Let L be a dense chain, L(µ,M) be an L-lattice, η, θ ⊆ µ such that η is an L-ideal of µ and θ is an L-dual ideal of µ. Then, η ∩ θ is an L-convex sublattice of µ if η ∩ θ ̸= O. Further, every L-convex sublattice of µ can be expressed in this form in one and only one way. Proof. Let η be an L-ideal of µ and θ be an L-dual ideal of µ. Then by Theorem 10, η and θ are L-convex sublattices of µ. Since intersection of L-convex sublattices of µ is an L-convex sublattice of µ, therefore η ∩ θ is an L-convex sublattice of µ provided η ∩ θ ̸= O (i.e., ∃ x ∈ M such that (η ∩ θ)(x) > 0). Next, let γ be an L-convex sublattice of µ. We take η = (γ]µ and θ = [γ)µ. We prove that γ = η ∩ θ. Clearly, γ ⊆ η and γ ⊆ θ. Therefore, γ ⊆ η ∩ θ. Suppose, γ ⊊ η ∩ θ. Then, ∃ x ∈ M such that γ(x) < η(x) ∧ θ(x). Let γ(x) = t. Then, η(x) > t and θ(x) > t. That is, x ∈ η>t and x ∈ θ>t . By Theorem 4, η>t is an ideal of µ> t and θ>t is a dual ideal of µ> t . This implies η>t = (η>t ] and θ>t = [θ>t ). Since η = (γ]µ, θ = [γ)µ and η(x) > t, θ(x) > t, therefore by Theorem 11, 12, ∃ r, s < a0, x ∈ (γ>r ], x ∈ [γ>s ), such that t < r and t < s. Thus we have, x ∈ (γ>r ] ⊆ (γ>r∧s], t < r; and x ∈ [γ>s ) ⊆ [γ>r∧s), t < s. That is, t ≤ r ∧ s. Moreover, A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 15 of 20 ∃ x1, . . ., xn ∈ γ>r∧s ⊆ µ> r∧s ∃ y1, . . ., ym ∈ γ>r∧s ⊆ µ> r∧s such that a = y1 ∧ . . . ∧ ym ≤ x ≤ x1 ∨ . . . ∨ xn = b. Since µ> r∧s is a sublattice of M and (γ>r∧s] is an ideal of µ> r∧s, we have a, b, x ∈ µ> r∧s. We also have a, b ∈ γ>r∧s as γ>r∧s is a sublattice of µ> r∧s. Now, γ being an L-convex sublattice of µ, γ>r∧s is a convex sublattice of µ> r∧s. Consequently, x ∈ γ>r∧s. That is, t = γ(x) > r∧s, which contradicts the fact that t < r and t < s (as L is a chain). Hence, γ(x) = (η ∩ θ)(x), ∀ x ∈ M. That is γ = η ∩ θ. Thus each convex sublattice γ of µ can be expressed in this form. To prove the uniqueness of this representation, suppose there exists an L-ideal η of µ and an L-dual ideal θ of µ such that γ = η∩θ. We prove that η = (γ]µ and θ = [γ)µ. Since γ ⊆ η, therefore (γ]µ ⊆ η. For reverse inclusion, let x ∈ M and η(x) = t. Then clearly, t ≤ a0. Since γ ⊆ η, therefore γt ⊆ ηt. If y ∈ γt , then x, y ∈ ηt. This implies x ∨ y ∈ ηt. We also have y ∈ γt ⊆ [γt), where [γt) is a dual ideal of µt and y ≤ x ∨ y in µt. Therefore, x ∨ y ∈ [γt) and t ≤ a0. This implies t ≤ [γ)(x ∨ y) = θ(x ∨ y). We also have η(x∨ y) ≥ t. Therefore, θ(x∨ y)∧ η(x∨ y) ≥ t. That is, γ(x∨ y) ≥ t. Thus, x ∨ y ∈ ηt ⊆ (γt]. Note that (γt] is an ideal of µt and x ≤ x ∨ y. Hence, x ∈ (γt] and t ≤ a0. Thus, η(x) = t ≤ ∨r≤a0{r : x ∈ (γr]} = (γ](x). Hence, η ⊆ (γ] and therefore, η = (γ]. Similarly, θ = [γ). Consequently, we get the uniqueness of the representation of γ. 4. Complement of an L-set and L-prime ideal and L-maximal ideal of an L-lattice In this section, the important concept of an order reversing involution on a lattice is discussed. Based on this notion, the complement of an L-lattice is defined. These notions occur frequently in Lattice Implication Algebras and L-topological spaces [11, 13, 16, 17]. If (L,≤,∧,∨) is a lattice, then L∗(= L) is also a lattice with respect to reverse order “≥”, where y ≥ x in L∗ if and only if x ≤ y in L. An order reversing involution on a lattice L is defined as a bijection τ : L → L∗ satisfying τ(τ(x)) = x, ∀ x ∈ L and x ≤ y in L if and only if τ(y) ≤ τ(x) in L = L∗. It is interesting to note that in all the examples provided in Section 3, there is an order reversing involution on the lattice L of truth values. The following result displays an inherent property of an order reversing involution. A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 16 of 20 Lemma 1. If L and L∗ are lattices and τ : L → L∗ is an order reversing bijection, then τ(a ∨ b) = τ(a) ∧ τ(b) and τ(a ∧ b) = τ(a) ∨ τ(b); ∀ a, b ∈ L. An order reversing involution defined on a lattice L of truth values leads to the definition of complement of an L-set as follows: Definition 10. Let µ ∈ LM and τ be an order reversing involution on L, i.e., τ : L → L∗ is a bijection satisfying τ(τ(x)) = x, ∀ x ∈ L and x ≤ y in L if and only if τ(y) ≤ τ(x) in L = L∗. Define an L-set µ′ : M∗ → L∗ as µ′(x) = τ(µ(x)), ∀ x ∈ M∗(= M). Then, µ′ ∈ LM and µ′ is called the complement of µ in LM . The following lemma establishes the De Morgan’s Laws in LM : Lemma 2. Let µ, η ∈ LM and τ be an order reversing involution on L. Then, (µ ∪ η)′ = µ′ ∩ η′ and (µ ∩ η)′ = µ′ ∪ η′. Proof. Let x ∈ M . Then, (µ ∪ η)′(x) = τ [(µ ∪ η)(x)] = τ [µ(x) ∨ η(x)] = τ(µ(x)) ∧ τ(η(x)) = µ′(x) ∧ η′(x) = (µ′ ∩ η′)(x). Hence, (µ ∪ η)′ = µ′ ∩ η′. The proof of the other part follows similarly. In the next result, it is proved that the complement of an L-prime ideal in M is an L-dual prime ideal in M . Theorem 14. Let τ be an order reversing involution on lattice L and µ be an L-prime ideal of M . Then µ′, the complement of µ in LM , is an L-dual prime ideal of M . Proof. Let x, y ∈ M . Since µ is an L-prime ideal of M , we have µ(x ∧ y) ≤ µ(x) ∨ µ(y). This implies, τ(µ(x ∧ y)) ≥ τ [µ(x) ∨ µ(y)] as τ is an order reversing involution. That is, µ′(x ∧ y) ≥ τ [µ(x) ∨ µ(y)] = τ(µ(x)) ∧ τ(µ(y)) = µ′(x) ∧ µ′(y). (1) A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 17 of 20 Further, if x ≤ y in M , then µ(x) ≥ µ(y) in L (as µ is an L-ideal of M). This implies τ(µ(x)) ≤ τ(µ(y)) in L∗. Thus, µ′(x) ≤ µ′(y) . (2) Moreover, as µ is an L-ideal of M , x ≤ x ∨ y and y ≤ x ∨ y in M implies µ(x) ≥ µ(x ∨ y) and µ(y) ≥ µ(x ∨ y). Thus, µ′(x) ≤ µ′(x ∨ y) and µ′(y) ≤ µ′(x ∨ y) and hence µ′(x) ∧ µ′(y) ≤ µ′(x) ≤ µ′(x ∨ y) . (3) By (1), (2) and (3), we get that µ′ is an L-dual ideal of M . To establish that µ′ is an L-dual prime ideal of M , note that µ(x ∨ y) ≥ µ(x) ∧ µ(y). This implies µ′(x ∨ y) = τ(µ(x ∨ y)) ≤ τ [µ(x) ∧ µ(y)] = τ(µ(x)) ∨ τ(µ(y)) = µ′(x) ∨ µ′(y). Consequently, µ′ is an L-dual prime ideal of M . By the above theorem, it can be concluded that if L is a lattice with an order reversing involution τ , then µ is an L-prime ideal of M if and only if µ′ is an L-dual prime ideal of M . The next theorem, combined with Theorem 14, leads to the following interesting analogue of a result from classical lattice theory: An L-ideal η of M is an L-prime ideal of M if and only if η′ is an L-dual ideal of M . In fact, η′ is an L-dual prime ideal of M . Theorem 15. Let τ be an order reversing involution on the lattice L and η be an L-ideal of M such that η′ is an L-dual ideal of M . Then, η and η′ are L-prime ideals of M . Proof. Suppose η is an L-ideal of M such that η′ is an L-dual ideal of M . We have η′(x ∧ y) ≥ η′(x) ∧ η′(y); ∀ x, y ∈ M. This implies η(x ∧ y) = τ [η′(x ∧ y)] ≤ τ [η′(x) ∧ η′(y)] = τ(η′(x)) ∨ τ(η′(y)) = η(x) ∨ η(y). A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 18 of 20 Thus, η is an L-prime ideal of M . Similarly, we have η(x ∧ y) ≥ η(x) ∧ η(y); ∀ x, y ∈ M. This implies η′(x ∧ y) = τ [η(x ∧ y)] ≤ τ [η(x) ∧ η(y)] = τ(η(x)) ∨ τ(η(y)) = η′(x) ∨ η′(y). Hence, η and η′ are L-prime ideals of M . In fact, in the above theorem, it can be proved that η′ is an L-dual prime ideal of M . We conclude this paper by discussing an analogue of another well known fact in classical lattice theory that, In a distributive lattice with the maximal element 1, every proper ideal is contained in a maximal ideal. The next theorem proves a similar result in an L-lattice µ. Theorem 16. Let L(µ,M) be an L-lattice, where L is a completely distributive lattice. Let η ⊆ µ be an L-ideal of µ. Then, there exists an L-maximal ideal θ of µ such that η ⊆ θ. Proof. Let I = {γ/γ is an L-ideal of µ such that η ⊆ γ}. Let Φ = {γi}i∈Λ be a chain in I. Then clearly, η ⊆ ∪{γi} and ∪ {γi} ⊆ µ. If x, y ∈ M , {∪γi}(x ∨ y) = ∨γi(x ∨ y) ≥ ∨[γi(x) ∧ γi(y)] (as each γi is an L-ideal of µ) = [∨γi(x)] ∧ [∨γi(y)] = (∪γi)(x) ∧ (∪γi)(y). Moreover, {∪γi}(x ∧ y) = ∨γi(x ∧ y) ≥ ∨[µ(x) ∧ γi(y)] (as each γi is an L-ideal of µ) = µ(x) ∧ [∨γi(y)] = µ(x) ∧ (∪γi)(y). A. Jain, I. Jahan / Eur. J. Pure Appl. Math, 18 (1) (2025), 5532 19 of 20 Thus, {∪γi} is an L-ideal of µ containing η. That is, {∪γi} ∈ I. Thus, every chain in I has an upper bound in I. Therefore, by Zorn’s Lemma, I has a maximal element. That is, ∃ a maximal L-ideal γ of µ such that η ⊆ γ. Hence, γ is the required L-maximal ideal of µ containing η. Conclusion In the present work, the concept of an L-convex sublattice in an L-lattice is studied in detail and the unique representation theorem for L-convex sublattices is established. Moreover, the concept of order reversing involution is utilized on the lattice L of truth values to define the complement of an L-set. The notion of complementation plays a significant role in the theory of Boolean Algebras, lattice implication Algebras and topo- logical spaces. In this paper, it is established that the concept of complementation of an L-set leads to proving some significant results in L-lattice theory. This notion is fur- ther worthy of attention as it may lead to some remarkable development in the theory of L-substructures of an L- Acknowledgements The authors are highly grateful to the learned referees for their valued comments which helped to improve the quality and presentation of this paper. References [1] N Ajmal and I Jahan. A study of normal fuzzy subgroups and characteristic fuzzy subgroup of a fuzzy group. Fuzzy Information and Engineering, 3:123–143, 2012. [2] N Ajmal and I Jahan. Nilpotency and theory of L subgroups of an L-group. 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