EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6418 ISSN 1307-5543 – ejpam.com Published by New York Business Global Amicable Sets in Paradistributive Latticoids Ramesh Sirisetti1, Jogarao Gunda2, Ravikumar Bandaru3, Thiti Gaketem4,∗ 1 Department of Mathematics, Aditya University, Surampalem, Kakinada, Andhra Pradesh 533437, India 2 Department of BS & H, Aditya Institute of Technology and Management, Tekkali, Srikakulam, Andhra Pradesh 530021, India 3 Department of Mathematics, School of Advanced Sciences, VIT-AP University, Andhra Pradesh 522237, India 4 Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand Abstract. In this work, we introduce maximal sets and amicable sets in a paradistributive latti- coid. We present a few number of examples and counter-examples for them, prove some algebraic properties on them. Also, we define center of a paradistributive latticoid and prove that it is equal to the intersection of all maximal sets. Finally, we obtain some necessary and sufficient conditions for a paradistributive latticoid to become relatively complemented. 2020 Mathematics Subject Classifications: 06D99, 06D20 Key Words and Phrases: Paradistributive latticoids, copatible sets, maximal sets, amicable sets, relatively complemented paradistributive latticoids 1. Introduction The exploration of algebraic structures such as lattices and their generalizations con- stitutes a fundamental area of study in universal algebra and mathematical logic. A recent advancement in this direction is the development of the theory of paradistributive latticoids (PDLs) [1], which extend classical distributive lattices by relaxing conventional distribu- tive constraints while maintaining essential structural properties. Introduced by Bandaru and Ajjarapu, PDLs are algebras of type (2, 2, 1), equipped with binary operations ∨ and ∧ and a designated greatest element 1, and they satisfy a set of axioms that emulate distributive behavior within a more generalized algebraic setting. This work builds upon the foundational framework of PDLs to introduce and system- atically investigate the concepts of compatible sets, maximal sets, and amicable sets. A ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6418 Email addresses: ramesh.sirisetti@gmail.com (R. Sirisetti), jogarao.gunda@gmail.com (J. Gunda), ravimaths83@gmail.com (R. Bandaru), thiti.ga@up.ac.th (T. Gaketem) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 2 of 13 compatible set is defined as a subset whose elements are mutually commutative under the operations ∨ and ∧, and a maximal set is characterized as a compatible set that is max- imal with respect to inclusion. The novel notion of amicability is introduced to describe maximal sets that exhibit a unifying interaction with the entire lattice, thereby revealing new algebraic insights. Through a rigorous methodology comprising formal definitions, illustrative examples, counterexamples, and theorems, this paper develops a comprehen- sive theory surrounding these structures. In particular, it establishes that the center of a PDL defined as the set of elements compatible with all elements of the lattice is precisely the intersection of all maximal sets, and furthermore, that this center forms a filter. The study of amicable sets serves as a key tool in the structural analysis of PDLs, leading to a new characterization of relatively complemented PDLs a property of considerable inter- est due to its connections with completeness and decomposability in lattice theory [2, 3]. The contributions of this work enhance the algebraic understanding of paradistributive latticoids by introducing and analyzing new invariants and internal structures. These re- sults extend earlier research on normal PDLs [4] and parapseudo-complementation [5], and are grounded in the classical algebraic foundations laid by Birkhoff [2] and Boole [6]. As such, this paper provides a substantive addition to the theoretical landscape of generalized lattice systems. 2. Preliminaries In this section, we present necessary definitions and properties of paradistributive latticoids which are taken from [1] for quick reference and to develop the theory. Definition 2.1. [1] An algebra (V,∨,∧, 1) of type (2, 2, 1) is called a Paradistributive Latticoid (Abbreviated as PDL), if it assures the following axioms: (i) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) (ii) (a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c) (iii) (a ∨ b) ∧ b = b (iv) (a ∨ b) ∧ a = a (v) a ∨ (a ∧ b) = a (vi) a ∨ 1 = 1, for all a, b, c ∈ V. Given a, b in a paradistributive latticoid V, we say that a is less than or equal to b and write a ≤ b, if a ∧ b = a or equivalently a ∨ b = b. It is observed that ≤ is a partial order on V and the element 1 is the greatest element. Throughout this paper, we refer V as a Paradistributive Latticoid with the greatest element 1. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 3 of 13 Example 2.2. [1] Let V be a non-empty set. Fix some element b0 ∈ V. For any a, b ∈ V, define ∨ and ∧ on V by a ∨ b = { a, if b ̸= b0 b0, if b = b0 and a ∧ b = { b, if b ̸= b0 a, if b = b0. Then (V,∨,∧, b0) is called a disconnected PDL, and b0 is the greatest element. Lemma 2.3. [1] For any a, b ∈ V, (i) a ∧ a = a (ii) a ∨ a = a (iii) (a ∧ b) ∨ b = b (iv) a ∨ (b ∧ a) = a (v) a ∧ (a ∨ b) = a. Lemma 2.4. [1] For any a, b, c, d ∈ V, (i) a ∧ 1 = a (ii) 1 ∧ a = a (iii) 1 ∨ a = 1 (iv) (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c) (v) a ∨ (b ∧ c) = a ∨ (c ∧ b) (vi) ∨ is associative in V (vii) d ∨ [a ∧ (b ∧ c)] = d ∨ [(a ∧ b) ∧ c] (viii) a ∨ (b ∨ c) = a ∨ (c ∨ b) (ix) a ∨ b = 1 ⇐⇒ b ∨ a = 1 (x) a ∨ b = 1 =⇒ b ∧ a = a ∧ b. Definition 2.5. [1] An element m ∈ V is said to be minimal, if for any a ∈ V,m ≤ a implies m = a. Lemma 2.6. [1] For any m ∈ V, the following are equivalent; (i) m is minimal, (ii) a ∧m = m, for all a ∈ V (iii) a ∨m = a, for all a ∈ V. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 4 of 13 A non-empty subset F of V is said to be a filter, if for any a, b ∈ F, x ∈ V, a ∧ b ∈ F and x ∨ a ∈ F . Let S be a non-empty subset of V. Then [S) = {a ∨ (∧n i=1si) | si ∈ S, a ∈ V, 1 ≤ i ≤ n and n is a positive integer } is the smallest filter of V containing S. If S = {a}, then we write [S) = [a), the principal filter generated by a. The collection F (V) of all filters of V forms a distributive lattice under set inclusion, in which, the glb and lub of any two filters F and G are given by F ∧G = F ∩G and F ∨G = {f ∧ g | f ∈ F and g ∈ G} respectively. Lemma 2.7. [1] Given a filter F of V and a, b ∈ V, (i) [a) = {x ∨ a | x ∈ V} (ii) a ∈ [b) ⇐⇒ a = a ∨ b (iii) a ∨ b ∈ F ⇐⇒ b ∨ a ∈ F (iv) [a ∨ b) = [b ∨ a) = [a) ∧ [b) (v) [a ∧ b) = [b ∧ a) = [a) ∨ [b). Lemma 2.8. [1] For any a, b ∈ V, (i) (a ∨ b) ∨ b = a ∨ b (ii) (a ∨ b) ∨ a = a ∨ b (iii) a ∨ (a ∨ b) = a ∨ b (iv) a ∧ (a ∧ b) = a ∧ b (v) (a ∧ b) ∧ b = a ∧ b (vi) b ∧ (a ∧ b) = a ∧ b. Theorem 2.9. [1] The following are equivalent in V; (i) (V,∨,∧, 1) is a distributive lattice (ii) The poset (V,≤) is directed below (iii) (a ∧ b) ∨ a = a, for all a ∈ V (iv) ∧ is commutative (v) ∨ is commutative (vi) The relation R = {(a, b) ∈ V× V | b ∨ a = b} is anti-symmetric. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 5 of 13 3. Amicable Sets in Paradistributive Latticoid In this section we define compatible sets and then maximal sets in a paradistributive latticoid and extensively study their algebraic properties, and provided a good number of counter-examples in each case. We define the center of a paradistributive latticoid using compatibility, and then we prove that the center is a filter and it is the intersection of all maximal sets in a paradistributive latticoid. Finally, we define amicable sets in a par- adistributive latticoid and then we characterize paradistributive latticoids using amicable sets. We obtain some necessary and sufficient conditions for a paradistributive latticoid to become relatively complemented. Definition 3.1. An element a ∈ V is said to be compatible with an element b ∈ V, if a ∨ b = b ∨ a or equivalently a ∧ b = b ∧ a, and it is denoted by a ∼ b. A subset S of V is said to be compatible, if a ∼ b, for all a, b ∈ S. Lemma 3.2. For any a, b, c ∈ V, we have the following; (i) 1 ∼ a (ii) ∼ is reflexive (iii) ∼ is symmetric (iv) a ≤ b =⇒ a ∼ b (v) a ∼ b =⇒ c ∨ a ∼ c ∨ b, a ∧ c ∼ b ∧ c, a ∨ c ∼ b ∨ c. Proof. Let a, b, c,∈ V. Then (i) a ∧ 1 = a = 1 ∧ a ⇒ a ∼ 1 (by Lemma 2.4((i) & (ii)). (ii) a ∧ a = a = a ∧ a ⇒ a ∼ a (by Lemma 2.3(i)). (iii) If a ∼ b, then a ∧ b = b ∧ a. Therefore b ∼ a. (iv) If a ≤ b, then a∧b = a. Now, b∧a = b∧(a∧b) = a∧b (By Lemma 2.8(vi)). Therefore a ∼ b. (v) Suppose a ∼ b. Then, (c ∨ a) ∧ (c ∨ b) =c ∨ (a ∧ b) (By Definition 2.1(i)) =c ∨ (b ∧ a) (By Lemma 2.4(v)) =(c ∨ b) ∧ (c ∨ a). (By Definition 2.1(i)) Therefore c ∨ a ∼ c ∨ b. Now, (a ∧ c) ∨ (b ∧ c) = (a ∨ b) ∧ c (By Lemma 2.4(iv)) = (b ∨ a) ∧ c (Since a ∼ b) = (b ∧ c) ∨ (a ∧ c). (By Lemma 2.4(iv)) Therefore a ∧ c ∼ b ∧ c. Similarly, (a ∨ c) ∧ (b ∨ c) = (a ∧ b) ∨ c (By Definition 2.1(ii)) = (b ∧ a) ∨ c (Since a ∼ b) = (b ∨ c) ∧ (a ∨ c). By Definition 2.1(ii)) R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 6 of 13 Therefore a ∨ c ∼ b ∨ c. Remark 3.3. Given a, b, c ∈ V, a ∼ b does not imply (c ∧ a) ∼ (c ∧ b). For, see the following example; Example 3.4. Let V = {a, b, c, d, 1} be a set with binary operations ∨ and ∧ given in the following; ∨ a 1 b c d a a 1 a c c 1 1 1 1 1 1 b b 1 b d d c c 1 c c c d d 1 d d d ∧ a 1 b c d a a a b a b 1 a 1 b c d b a b b a b c a c b c d d a d b c d Then (V,∨,∧, 1) is a Paradistributive Latticoid, and 1 ∼ b but (c∧ 1)∨ (c∧ b) ̸= (c∧ b)∨ (c ∧ 1)). Theorem 3.5. V is distributive if and only if a ∼ b implies c∧a ∼ c∧b, for all a, b, c ∈ V. Proof. If V is distributive, then it is easy to verify the result. On the other hand, let a, b ∈ L. By our assumption, b ∧ 1 ∼ b ∧ a (that is (b ∧ 1) ∨ (b ∧ a) = (b ∧ a) ∨ (b ∧ 1)). So that b ∨ (b ∧ a) = (b ∧ a) ∨ b. By Definition 2.1(v)., b = (b ∧ a) ∨ b. Therefore V is a distributive lattice (By Theorem 2.9.). Example 3.6. Let I be a non-empty set and V is a disconnected PDL. Then (VI ,∨,∧, 1) is a PDL with respect to the point wise operations, where 1̄ : I → V defined by 1̄(i) = 1 for all i ∈ I, is the greatest element in V. In this V, for any x, y ∈ V, we have (i) x ∼ y if and only if x||x|∩|y| = y||x|∩|y| and (ii) x ≤ y if and only if x|x| = y|y|, where |x| = {i ∈ I|x(i) ̸= 1}. Example 3.7. Let R be a commutative regular ring with unity [1]. For any a, b ∈ R, a ∼ b if and only if ab2 = a2b. For any a, b ∈ V, a ∼ b ⇒ a ∨ b = b ∨ a ⇒ b0a = a0b (a ∨ b = b0a) ⇒ bb0a = ba0b ⇒ ba = b2a0 (bb0 = b) ⇒ baa = b2a0a ⇒ ba2 = b2a (a0a = a) ⇒ a2b = ab2. (commutative) Now, a2b = ab2 ⇒ ab(a− b) = 0 ⇒ a0b0(a− b) = 0 (aR = a0R) ⇒ a0b0a− a0b0b = 0 ⇒ b0a0a = a0b (bb0 = b) ⇒ b0a = a0b (aa0 = a) ⇒ a ∧ b = b ∧ a. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 7 of 13 Therefore a ∼ b. Definition 3.8. A maximal compatible set in V is called a maximal set. Example 3.9. In the Example 2.2., {x, 1} is a maximal set, for any x ∈ V and x ̸= 1. Example 3.10. Let V be a disconnected PDL, I an infinite set and L = {x ∈ VI | |x| is finite }, where |x| = {i ∈ I | x(i) ̸= 1}. Then (L,∨,∧, 1) is a PDL with respect to the point wise operations, where 1̄ : I → V, defined by 1̄(i) = 1, for all i ∈ I. In this L, we observe the following; (i) L has no minimal elements Let x ∈ L. Then |x| is finite. Choose i0 ∈ I such that i0 /∈ |x| and fix 1 ̸= d ∈ V. Now, for i ∈ I, define, y(i) =  x(i), if i ∈ |x| d, if i = i0 1, otherwise. Then y ∈ L and y ≤ x. Therefore L has no minimal element. (ii) If p ∈ (V− {0})I , then the set Mp = {pS | S is a finite subset of I} is a maximal set in L, where pS : I → V is defined by pS(i) = { p(i), if i ∈ S 1, if i /∈ S, for all i ∈ I. Let S1, S2 be two finite subsets of I and i ∈ S1 ∩ S2. Now, (pS1 ∧ pS2)(i) ⇒ pS1(i) ∧ pS2(i) ⇒ p(i) ∧ p(i) ⇒ pS2(i) ∧ pS1(i) ⇒ (pS2 ∧ pS1)(i). If i /∈ S1 ∩ S2, then (pS1 ∧ pS2)(i) = pS1(i) ∧ pS2(i) = 1. Therefore pS1 ∧ pS2 = pS2 ∧ pS1 , and hence Mp is a compatible set. Let x ∈ L such that x ∼ pS for all finite subset S of I. Since |x| is a finite subset of I, we have x ∼ p|x|. Therefore x|x| = p|x| and hence x = p|x| ∈ Mp. Thus Mp is a maximal set. Example 3.11. The set Mm = {a ∈ V | m ≤ a}, where m is a minimal element in V, is a maximal set in V. Let m be a minimal element in V. Then m ∈ Mm. Therefore Mm is non-empty. Let a, b ∈ Mm. Then m ≤ a, b. Therefore m∧a = m∧b = m. Now, a∧b = (m∨a)∧ (m∨b) = m∨ (a∧ b) = m∨ (b∧a) = (m∨ b)∧ (m∨a) = b∧a. Therefore a∧ b = b∧a. So that a ∼ b. Hence Mm is a compatible set. Assume that S is a compatible set such that Mm ⊆ S. Let c ∈ S. Then c ∼ s, for all s ∈ S. Since Mm ⊆ S, a ∼ m. Now, m = a∧m = m∧ a. Then m ≤ a. Therefore a ∈ Mm. So that S ⊆ Mm. Hence Mm = S. Thus Mm is a maximal set in V. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 8 of 13 Let {Vi}i∈∆ be a family of PDLs. Then it is easy to observe that the product ∏ i∈∆Vi with point wise operations is a PDL. Now, we have the following; Theorem 3.12. For any non-empty subset M of ∏ i∈∆Vi, M is maximal set in ∏ i∈∆Vi if and only if M = ∏ i∈∆Mi, where Mi is a maximal set in Vi. Proof. Let {Vi}i∈∆ be the family of PDLs. If Mi is a maximal set in Vi. Then clearly∏ i∈∆Mi is a maximal set. On the other hand, let M be the set such that Mi = {a(i) | a ∈ M}, then Mi is non-empty and compatible set in Vi. Let b ∈ Vi is such that b ∼ a(i), for all a ∈ M . Now, define c ∈ ∏ j∈∆Vj by c(j) = b, if i = j and 1j if i ̸= j. Now, d ∈ M and j ∈ ∆, (d ∧ c)(j) = d(j) ∧ c(j) = c(j) ∧ d(j) = (c ∧ d)(j). Therefore c ∼ d, for all d ∈ M . So that c ∈ M and c(j) = b ∈ Mi. Hence Mi is a maximal set in Vi for all i ∈ ∆. Since∏ i∈∆Mi is a compatible set containing M . Therefore M = ∏ i∈∆Mi. Lemma 3.13. Let M be a maximal set in V and c ∈ V such that c ∼ a, for all a ∈ M . Then c ∈ M . Proof. Let M ′ = M∪{c}. Then M ′ is compatible set and M ⊆ M ′. By the maximality of M , M = M ′ and hence c ∈ M . Theorem 3.14. Let M be a maximal set in V. Then we have (i) M contains 1 (ii) M is distributive lattice with greatest element 1 (iii) M is an initial segment. Proof. (i) Let a ∈ M . Then a ∧ 1 = a = 1 ∧ a. Therefore 1 ∼ a, for all a ∈ M . By Lemma 3.13., 1 ∈ M . (ii) Let a, b ∈ M . Then a ∼ b. Now, a ∧ (a ∧ b) = a ∧ b and (a∧ b)∧ a = (b∧ a)∧ a = b∧ a = a∧ b. Therefore a ∼ a∧ b. By Lemma 3.13., a∧ b ∈ M. Now, a ∧ (a ∨ b) = a and (a ∨ b) ∧ a = (b ∨ a) ∧ a = a. So that M is closed under ∨ and ∧. By Theorem 2.9., M is a distributive lattice with greatest element 1. (iii) Let a ∈ M and b ∈ V. If b ≤ a, then b ∧ a = b. Now, a ∧ b = a ∧ (b ∧ a) = b ∧ a (By Lemma 2.8(vi)). Therefore a ∼ b, for all a ∈ M . By Lemma 3.13., b ∈ M . Hence M is an initial segment. Definition 3.15. In a PDL V, we define the center of V as the set C(V) = {a ∈ V | a ∼ c, for all c ∈ V }. Theorem 3.16. C(V) is the intersection of all maximal sets in V. Proof. If a ∈ C(V) and M is a maximal set in V. Then M ∪ {a} is compatible and by the maximality of M , a ∈ M (By Lemma 3.13.). Therefore C(V) ⊆ M , for all maximal sets in V. On the other hand, let a ∈ V such that a /∈ C(V). Then there exists c in V such that a is not compatible with c. Put P = {C ⊆ V | C is compatible, c ∈ C and a /∈ C}. Then {c} ∈ P. Therefore P is non-empty and (P,⊆) is a partial order set. Let R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 9 of 13 C1 ⊆ C2 ⊆ C3 ⊆ ... ⊆ Cn ⊆ .... be a chain in P. If a ∈ ∪i∈ICi. Then a ∈ Ci, for some i which is a contradiction to a /∈ Ci, for every i. If ∪i∈ICi is not comparable, then there exist b, c ∈ ∪i∈ICi such that b is incomparable with c where b ∈ Ci, c ∈ Cj , for some i, j ∈ I. But Ci ⊆ Cj or Cj ⊆ Ci, b, c ∈ Ci or b, c ∈ Cj . Then b ∼ c which is contradiction. So that ∪i∈ICi is compatible. Therefore ∪i∈ICi is an upper bound in P. By Zorn’s lemma, P has a maximal member, say Q. Suppose N is compatible in V such that Q ⊂ N. Then c ∈ N. Since c is incomparable a, a /∈ N. Therefore N ∈ P. By maximality of Q in P, we get Q = N. Thus Q is maximal set in V and a /∈ Q. This proves that C(V) is equal to the intersection of all maximal sets in V. Theorem 3.17. C(V) is a filter of V. Proof. It is easy to observe that C(V) is the intersection of all maximal sets in V. Since each maximal set is closed under ∧ and ∨, we have C(V) is closed under ∧ and ∨. Let a ∈ C(V), b, c ∈ V. Then, c ∨ (b ∨ a) = (c ∨ b) ∨ a (∨ is associative) = a ∨ (c ∨ b) (since a ∈ C(V)) = a ∨ (b ∨ c) (By Lemma 2.4(viii)) = (a ∨ b) ∨ c (∨ is associative) = (b ∨ a) ∨ c (since a ∈ C(V)) (b ∨ a) ∨ c = c ∨ (b ∨ a). Therefore b ∨ a ∈ C(V), for all b ∈ V. Thus C(V) is a filter of V. Theorem 3.18. If {Vα}α∈∆ is a family of PDLs, then C( ∏ α∈∆Vα) = ∏ α∈∆C(Vα). Proof. Let a ∈ C( ∏ β∈∆Vβ). Then a ∼ x, for all x ∈ ∏ β∈∆Vβ. Let α ∈ ∆ and s ∈ Vα. If x ∈ ∏ β∈∆Vβ) such that x(α) = s. Since a ∈ C( ∏ β∈∆Vβ), a ∼ x and hence a ∧ x = x ∧ a. Therefore (a ∧ x(α)) = a(α ∧ x(α) = x(α) ∧ a(α) = s ∧ a(α). This is true for all s ∈ Vα. So that a(α) ∈ C(Vα), for all α ∈ ∆. Thus a ∈ ∏ α∈∆C(Vα). On the other hand, let a ∈ ∏ α∈∆C(Vα). Then a(α) ∈ C(Vα), for all α ∈ ∆. Let x ∈ ∏ α∈∆Vα. Then x(α) ∈ Vα, for all α ∈ ∆. Since a(α) ∈ C(Vα), for all α ∈ ∆, a(α) ∼ x(α), for all α ∈ ∆. Therefore a ∼ x and hence a ∈ C( ∏ α∈∆Vα). Thus C( ∏ α∈∆Vα) = ∏ α∈∆C(Vα). Definition 3.19. A maximal set M in V is said to be amicable with an element a of V, if there exists b ∈ M such that a ∨ b = a. The set M is said to be amicable if it is amicable with every element of V. Example 3.20. In Example 2.2., for any x ̸= 1, {x, 1} is an amicable set . Example 3.21. Let V be a PDL with minimal elements. Then every amicable set is of the form Mm = {x ∈ V | x ≤ m}, where m is a minimal element in V. Theorem 3.22. Let M be a amicable set in V. Then for each a ∈ V, there exists a unique a0 ∈ M such that [a0) = [a) i.e., a0 ∨ a = a0 and a ∨ a0 = a. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 10 of 13 Proof. Let a ∈ ∨. Then there exists b ∈ M such that a∨ b = a. Take a0 = b∨ a. Now, a0∨b = (b∨a)∨b = b∨(a∨b = b∨a = a0 (By Lemma 2.4(viii) and b∨a0 = b∨b∨a = b∨a = a0. Therefore a0 ∼ b. Since b ∈ M, a0 ∈ M . Now, a0 ∨ a = b ∨ a ∨ a = b ∨ a = a0. Then [a0) ⊆ [a). Therefore a ∈ [a0). Now, a ∨ a = a ∨ b ∨ a = a ∨ a ∨ b = a ∨ b = a (By Lemma 2.8(iii)). Then [a) ⊆ [a0). Therefore a ∈ [a0). Hence [a0) = [a). Suppose there exists c ∈ M such that [c) = [a) = [a0). Since a0, c ∈ M, a0 ∼ c and hence c = c ∨ a0 = a0 ∨ c. Therefore a0 ≤ c. Since a0 ∈ [c), a0 = a0 ∨ c = c ∨ a0 and hence c ≤ a0. Thus a0 = c. Let us denote η = {(a, b) ∈ V× V | [a) = [b)}. Then we have the following Theorem 3.23. η is a congruence relation on V. Proof. Let a, b, c ∈ V. (i) Reflexive: For any a ∈ V, [a) = [a) ⇔ (a, a) ∈ η. (ii) Transitive: For any (a, b) ∈ η and (b, c) ∈ η, [a) = [b) = [c). Therefore [a) = [c). Hence (a, c) ∈ η. (iii) Symmetric: (a, b) ∈ η ⇔ [a) = [b) ⇔ (b, a) ∈ η. Therefore η is an equivalence relation on V. Let (a, b), (c, d) ∈ η. Then [a∨ c) = [a)∧ [c) = [b)∧ [d) = [b∨d) and [a ∧ c) = [a) ∨ [c) = [b) ∨ [d) = [b ∧ d). Therefore (a ∨ c, b ∨ d), (a ∧ c, b ∧ d) ∈ η. Hence η is a congruence relation on V. Lemma 3.24. For any element a ∈ V, a/η = {1} if and only if a = 1. Proof. Let a ∈ V. Then a ∈ a/η. Suppose that a/η = {1}. Then [a) = [1). Therefore a = a ∨ 1 = 1. Hence a = 1. On the other hand, let a = 1, 1/η = {a ∈ V | (a, 1) ∈ η} = {a ∈ V | [a) = [1)} = {a ∈ V | a = a ∨ 1 = 1} = {1}. Hence a/η = {1}. Theorem 3.25. The quotient lattice V/η forms a distributive lattice with the least element 1/η = {1} and the operations a/η ∧ b/η = (a ∧ b)/η and a/η ∨ b/η = (a ∨ b)/η. It is easy to observe that η is the smallest congruence on V such that V/η is a dis- tributive lattice. Theorem 3.26. A compatible set M in V is amicable if and only if M ∩(a/η) is singleton set, for all a ∈ V. Proof. Suppose that M is amicable. Let a ∈ V. By Theorem 3.22., there exists a0 ∈ M such that [a0) = [a). Then a0 ∈ M ∩ (a/η). Now, b ∈ M ∩ (a/η) ⇒ (a, b) ∈ η and b ∈ M ⇒ [a) = [b) and b ∈ M ⇒ [a0) = [b) and b ∈ M (since [a0) = [a)) ⇒ b = a0. (since M is distributive lattice) Therefore M ∩ (a/η) = {a0}. Conversely suppose that M ∩ (a/η) is a singleton, for all a ∈ V. Let b ∈ V such that b ∼ a, for all a ∈ M . Then M ∩ (b/η) = {c}, for some c ∈ M(By our assumption). Then [b) = [c). Since b ∼ c, b = c ∈ M . Hence M is maximal set. Now, we show that M is amicable. Let a ∈ V. By our assumption M ∩ (a/η) = {d}, for some d ∈ M . Then [a) = [d). Therefore a = a ∨ d and d ∈ M . Hence M is amicable. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 11 of 13 Theorem 3.27. Let M be an amicable set in V and x ∈ V. Then Mx = {(a∧x)∨a | a ∈ M} is a compatible set. Proof. Let a, b ∈ M . Then [(b ∧ x) ∨ b] ∨ [(a ∧ x) ∨ a] ⇒ [(b ∧ x) ∨ b] ∨ [a ∨ (a ∧ x)] (By Lemma 2.4(viii)) ⇒ [(b ∧ x) ∨ b] ∨ a (By Definition 2.1(v)) ⇒ (b ∧ x) ∨ (b ∨ a) (∨ is associative) ⇒ (b ∨ (b ∨ a)] ∧ (x ∨ (b ∨ a)] (By Definition2.1(ii)) ⇒ (a ∨ b) ∧ (x ∨ (a ∨ b)] (since a ∼ b, a, b ∈ M) ⇒ [a ∨ (a ∨ b)] ∧ [x ∨ (a ∨ b)] (By Lemma2.8(iii)) ⇒ (a ∧ x) ∨ (a ∨ b) ⇒ [(a ∧ x) ∨ a] ∨ b (∨ is associative) ⇒ [(a ∧ x) ∨ a] ∨ [b ∨ (b ∧ x)] (By Definition 2.1(v)) ⇒ [(a ∧ x) ∨ a] ∨ [(b ∧ x) ∨ b] (By Lemma 2.4(viii) Therefore Mx is compatible set. Theorem 3.28. Let M be an amicable set in V. Then M is isomorphic to V/η. Proof. Define a map f : M → V/η by a/η, for all a ∈ M . Now, f(a∧ b) = (a∧ b)/η = a/η ∧ b/η = f(a) ∧ f(b) and f(a ∨ b) = (a ∨ b)/η = a/η ∨ b/η = f(a) ∨ f(b), for all a, b ∈ M . Then f is a homomorphism. Now, f(a) = f(b) ⇒ a/η = b/η. Let x ∈ a/η. Then (a, x), (b, x) ∈ η. Therefore [a) = [x) = [b). Therefore a = b. So that f is one- one. By Theorem 3.22., for x ∈ V, there exists x0 ∈ M such that [x0) = [x). Therefore x0/η = x/η = f(x0). So that f is onto and hence f is bijective. Theorem 3.29. Let M be an amicable set in V. Then the lattice of filters of V is iso- morphic with the lattice of filters of M . Proof. Let F (V) and F (M) are the lattices of filters of V and filters of M respectively. Define a map g : F (V) → F (M) by g(F ) = F ∩M , for all F ∈ F (V). Then clearly g is an order preserving map. Suppose F,G ∈ F (V) and F ∩M ⊆ G ∩M . For x ∈ F , there exists x0 ∈ M such that [x0) = [x). Therefore x0 ∈ F and hence x0 ∈ F ∩M ⊆ G ∩M . For this x0 ∈ G ∩ M, x ∈ [x) = [x0) ⊆ G ∩ M . Therefore x ∈ G. So that F ⊆ G. Hence g is one-one. Let K ∈ F (M). Put Q = {x ∈ V | x0 ∈ K such that [x) = [x0)}. Let x, y ∈ Q. Then there exist x0, y0 ∈ K such that [x) = [x0) and [y) = [y0). Now, [x∧y) = [x)∨[y) = [x0)∨[y0)) = [x0∧y0) and [x∨y) = [x)∧[y) = [x0)∧[y0)) = [x0∨y0)(since K is a filter). Let a ∈ V. Then there exists a0 ∈ M such that [a) = [a0). Now, [x∨ a) = [x)∧ [a) = [x0)∧ [a0) = [x0 ∨ a0). Since x0 ∨ a0 ∈ M , x∨ a ∈ F and hence Q is a filter of V and g(Q) = Q ∩M = K. Hence g is bijective and g−1 is also order preserving. Thus g is isomorphic. Definition 3.30. A PDL (V,∨,∧, 1) is said to be relatively complemented, if every interval [a, b], in V is a complemented lattice, for all a, b ∈ V. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 12 of 13 Theorem 3.31. V is relatively complemented if and only if every maximal set is relatively complemented. Proof. Suppose V is relatively complemented and M is a maximal set in V. For any a, b ∈ M with a ≤ b, the interval [a, b] in V is a subset of M . Let x ∈ [a, b]. Then a ≤ x. Therefore a ∧ x = x ∧ a and a ∈ M . So that x ∈ M . Hence [a, b] ⊆ M . Since V is relatively complemented, [a, b] is relatively complemented. Hence M is relatively complemented. Conversely suppose that every maximal set is relatively complemented. Let a, b ∈ V. Then [a, b] is compatible and it contained in a maximal set(since every maximal set is compatible). Therefore [a, b] is relatively complemented. By Definition 3.30., V is relatively complemented. Theorem 3.32. V is relatively complemented if and only if every amicable is relatively complemented. Proof. Let M be an amicable set in V such that M is relatively complemented. Let a, b ∈ V and a ≤ b. Then there exists a0, b0 ∈ M such that [a0) = [a) and [b0) = [b) (By Theorem 3.22.) Now, a0 ∨ b0 = b0 ∨ a0 (since a0, b0 ∈ M) = b0 ∨ a0 ∨ a (since a0 ∈ [a)) = b0 ∨ a ∨ a0 (By Lemma 2.4(viii)) = b0 ∨ a (since a ∈ [a0)) = b0 ∨ b ∨ a (since b0 ∈ [b)) = b0 ∨ b (since a ≤ b) = b0. (since b0 ∈ [b)) Therefore a0 ∧ b0 = a0 and hence a0 ≤ b0. Since M is relatively complemented, there exists x ∈ M such that a0 ∨ x = 1 and a0 ∧ x is minimal. For x ∨ b ∈ V, a ∨ (x ∨ b) = a ∨ a0 ∨ x ∨ b (since a ∈ [a0) = a ∨ 1 ∨ b (since a0 ∨ x = 1) = 1 a ∧ (x ∨ b) = (a ∨ a0) ∧ (x ∨ b) (since a ∈ [a0) = {a ∧ (x ∨ b)} ∨ {a0 ∧ (x ∨ b)} (By Lemma 2.4(iv)) = {a ∧ (x ∨ b)} ∨ {(x ∨ b) ∧ a0} (By Lemma 2.4(v)) = (x ∨ b) ∧ {[a ∧ (x ∨ b)] ∨ a0} (By Lemma 2.3(iii)) = (x ∨ b) ∧ {(a ∨ a0) ∧ [(x ∨ b) ∨ a0]} (By Definition 2.1(ii)) = (x ∨ b) ∧ {(a ∨ a0) ∧ [(x ∨ a0 ∨ b]} (By Lemma 2.4(vi)(viii)) = (x ∨ b) ∧ (a ∨ a0) (since a0 ∨ x = 1) = (x ∨ b) ∧ a (since a ∈ [a0) and a0 ∈ [a) = (x ∧ a) ∨ (b ∧ a) (By Lemma 2.4(iv)) = (x ∧ a) ∨ a (since a ≤ b)) = a. Therefore a ≤ x∨ b ≤ 1. So that a∨ (x∨ b) = 1 and a∧ (x∨ b) = a. Hence V is relatively complemented. R. Sirisetti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6418 13 of 13 4. Conclusion and Future Work In this paper, we introduced and analyzed maximal and amicable sets in paradistribu- tive latticoids (PDLs). We established key structural properties, including that the center of a PDL is the intersection of all maximal sets and forms a filter. Additionally, we charac- terized relatively complemented PDLs using amicable sets. These results contribute to the deeper understanding of compatibility in generalized lattices. Future work may focus on extending these concepts to related algebraic structures, exploring categorical properties, and developing computational approaches for identifying maximal and amicable sets in practical and theoretical applications. 5. Conflict of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper. Acknowledgements This research was supported by University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2026, Grant No. 2287/2568) References [1] Ravikumar Bandaru and Suryavardhani Ajjarapu. Paradistributive latticoids. Euro- pean Journal of Pure and Applied Mathematics, 17:819–834, 2024. [2] G. Birkhoff. Lattice theory. Amer. Math. Soc. Collequium Pub, 1967. [3] Y. L. Ershov. Relatively complemented distributive lattices. Algebra and Logic, 18:431– 459, 1978. [4] Ravikumar Bandaru, Prashant Patel, Noorbhasha Rafi, Rahul Shukla, and Suryavard- hani Ajjarapu. Normal paradistributive latticoids. European Journal of Pure and Applied Mathematics, 17:1306–1320, 2024. [5] Suryavardhani Ajjarapu, Ravikumar Bandaru, Rahul Shukla, and Young Bae Jun. Parapseudo-complementation on paradistributive latticoids. European Journal of Pure and Applied Mathematics, 17:1129–1145, 2024. [6] G. Boole. An investigation of the laws of thought. Reprinted by Open Court Publishing Co., Chelsea, London, 1940. Originally published in 1854. Introduction Preliminaries Amicable Sets in Paradistributive Latticoid Conclusion and Future Work Conflict of Interest