EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1129-1145 ISSN 1307-5543 – ejpam.com Published by New York Business Global Parapseudo-complementation on Paradistributive Latticoids Suryavardhani Ajjarapu1, Ravikumar Bandaru2, Rahul Shukla3,∗, Young Bae Jun4 1 Department of Mathematics, GITAM Deemed to be University, Hyderabad Campus, Telangana-502329, India 2 Department of Mathematics, School of Advanced Sciences, VIT-AP University, Andhra Pradesh-522237, India 3 Department of Mathematical Sciences and Computing, Walter Sisulu University, Mthatha 5117, South Africa 4 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea Abstract. In this paper, we introduce the concept of a parapseudo-complementation in a par- adistributive latticoid(PDL) and investigate its elementary properties. We demonstrate the inde- pendence of the axioms related to its definition, highlighting the flexibility of this concept. Addi- tionally, we establish necessary conditions for a PDL with a minimal element to be parapseudo- complemented and explore the properties required for parapseudo-complementation to be equation- ally definable. Moreover, we establish a one-to-one correspondence between the set of all minimal elements and the set of all parapseudo-complementations. 2020 Mathematics Subject Classifications: 06D99 Key Words and Phrases: Parapseudo-complementation, Paradistributive Latticoid(PDL), Min- imal element, Filter, Boolean algebra. 1. Introduction In the realm of algebraic structures, a variety of algebras, including lattices and Boolean algebras, provide generalizations of the concept of complement. Within this context, the notion of pseudo-complementation has been extended to encompass a wide range of semigroups, referred to as pseudo-complemented semilattices. The study of pseudo- complements in distributive lattices was first introduced and extensively researched by G. Birkhoff[2] and Orrin Frink[5]. I. Chajda et al.[3, 4] introduced the so-called sectionally ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5042 Email addresses: syerrapr@gitam.in (S. Ajjarapu), ravimaths83@gmail.com (R. Bandaru), rshukla@wsu.ac.za (R. Shukla), skywine@gmail.com (Y. B. Jun) https://www.ejpam.com 1129 © 2024 EJPAM All rights reserved. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1130 pseudocomplemented lattices and posets and demonstrated their roles in algebraic struc- tures. They defined congruences and filters in their structures, derived mutual relationship between them and described basic properties of congruences in strongly sectionally pseu- docomplemented posets. Later, the concept of a relative pseudocomplemented lattice was introduced by R. P. Dilworth(Dilworth 1939) where he interpreted relative pseudocom- plement as logical connective implication. M. Mandelker[6] expanded on this concept by introducing and investigating the notions of relative annihilators in lattices and relatively pseudo-complemented lattices. Mandelker proposed the annihilator (a, b) of element a relative to b as a natural generalization of the pseudo-complement a ∗ b. It represents the set of elements x satisfying a ∩ x ≤ b. The greatest element of (a, b), if it exists, is defined as the relative pseudo-complement a ∗ b. Thus, a lattice is considered relatively pseudo-complemented if each annihilator has a greatest element, making it a principal ideal. A dual weakly complemented lattice was introduced by Wille[15] and Kwuida[13]. Their contributions connected to the notion of annihilators of distributive dual weakly complemented lattice with a certain type of ideals called as closed ideals and later proved that closed ideals depend on the dual weak complementation operation on the lattice of all ideals I(L) of L. Eman Ghareeb Rezk[9] introduced the concept of closed ideals and annihilators over the class of distributive dual weakly complemented lattices. The con- nection between closed ideals and annihilators in this class was obtained. M. S. Rao[7] introduced the concept of δ-ideals in pseudo-complemented distributive lattices and then Stone lattices are characterized in terms of δ-ideals. Further the properties of normal ideals of pseudo-complemented distributive lattices and the characterization of disjunctive lattices with the help of normal ideals was studied by M. S. Rao et al.[8] The theory of pseudo-complements for posets was developed by P. V. Venkatanarasimhan [14], who introduced the concepts of ideals and semi-ideals and derived several results that paved the way for research on pseudo-complements in distributive lattices. These findings revealed that if every element in a pseudo-complemented semilattice or dual semilattice is normal, the algebra can be classified as a Boolean algebra. This conclusion led to new proofs for well-known theorems, such as the existence of maximal ideals in posets and the product of all maximal dual ideals being the dual ideal of dense components in a poset with a zero element. U. M. Swamy and G. C. Rao[11] introduced the concept of an Almost Distributive Lattice (ADL) as a unifying abstraction for various lattice-theoretic generalizations of Boolean algebras and Boolean rings. Furthermore, in collaboration with G. N. Rao[12], they extended the concept of pseudo-complementation to almost distributive lattices and demonstrated that the class of pseudo-complemented ADLs is equationally definable. They also explored the relationship between annihilator ideals and pseudo-complementations in an ADL and established a one-to-one correspondence between pseudo-complementations and maximal elements in an ADL assuming the existence of one pseudo-complementation. Recently, R. K. Bandaru et al.[1] introduced the concept of a Paradistributive Latti- coid(PDL) as a generalization of distributive lattice and investigated its properties. They introduced the notions of an ideal and a filter in a PDL and studied their properties. They proved a subdirect representation theorem for associative PDLs which simplifies R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1131 many results in PDLs. The main objective of this paper is to introduce the concept of parapseudo-complementation in a Paradistributive Latticoid (PDL) and investigate its properties. We provide exam- ples to illustrate the independence of the axioms defined for parapseudo-complementation. Specifically, we prove that a PDL V is parapseudo-complemented if and only if the anni- hilator filter [ρ]• is a principal filter for any ρ ∈ V . Additionally, we establish a one-to-one correspondence between the set of all minimal elements and the set of all parapseudo- complementations in V . Finally, we demonstrate that the corresponding Boolean algebras V ♦ and V ♢ are isomorphic. The remainder of this paper is structured as follows: section 1 provides a brief intro- duction to the concept of pseudo-complementation, followed by preliminaries in section 2. Section 3 presents the definition of parapseudo-complementation on a PDL, highlighting the independence of the axioms through illustrative examples. In section 4, we delve into the heart of our investigation by proving the necessary and sufficient conditions for a Par- adistributive Latticoid (PDL) with a minimal element to be parapseudo-complemented. Additionally, we establish that the class of parapseudo-complemented PDLs is equationally definable, providing a solid foundation for further exploration of this concept. Moving forward to section 5, we focus on the independence of the parapseudo-complementation ♦ within the corresponding Boolean algebra V ♦. By presenting a rigorous proof, we demon- strate that the structure and properties of V ♦ are not affected by the specific choice of parapseudo-complementation. This insight enhances our understanding of the relationship between parapseudo-complementation and the underlying Boolean algebra. In summary, through our research, we establish the necessary and sufficient conditions for parapseudo-complementation in PDLs, highlight the equationally definable nature of parapseudo-complemented PDLs, and demonstrate the independence of the parapseudo- complementation within the associated Boolean algebra. This study contributes to a deeper understanding of parapseudo-complementation and its implications in the context of Paradistributive Latticoids. 2. Preliminaries First we recall the necessary definitions and results from [1]. Definition 1. An algebra (V,∨,∧, 1) of type (2,2,0) is called a Paradistributive Latticoid, abbreviated as PDL, if it assures the subsequent axioms: (LD∨) κ1 ∨ (κ2 ∧ κ3) = (κ1 ∨ κ2) ∧ (κ1 ∨ κ3). (RD∨) (κ1 ∧ κ2) ∨ κ3 = (κ1 ∨ κ3) ∧ (κ2 ∨ κ3). (L1) (κ1 ∨ κ2) ∧ κ2 = κ2. (L2) (κ1 ∨ κ2) ∧ κ1 = κ1. (L3) κ1 ∨ (κ1 ∧ κ2) = κ1. (I1) κ1 ∨ 1 = 1. for any κ1, κ2, κ3 ∈ V . For any κ1, κ2 ∈ V , we say that κ1 is less than or equal to κ2 and write κ1 ≤ κ2 if R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1132 κ1 ∧ κ2 = κ1 or equivalently κ1 ∨ κ2 = κ2 and it can be easily observed that ≤ is a partial order on V . The element 1, in Definition 1, is called the greatest element. Example 1. Let V be a non-empty set. Fix some element ϱ0 ∈ V . Then, for any ρ, ϱ ∈ V define ∨ and ∧ on V by ρ ∨ ϱ = { ρ ϱ ̸= ϱ0 ϱ0 ϱ = ϱ0 and ρ ∧ ϱ = { ϱ ϱ ̸= ϱ0 ρ ϱ = ϱ0 Then (V,∨,∧, ϱ0) is a disconnected PDL with ϱ0 as its greatest element. Lemma 1. Let (V,∨,∧, 1) be a PDL. Then for any κ1, κ2, κ3, κ4 ∈ V , we have the follow- ing: (1) 1 ∧ κ1 = κ1. (2) κ1 ∧ 1 = κ1. (3) 1 ∨ κ1 = 1. (4) (κ1 ∨ κ2) ∧ κ3 = (κ1 ∧ κ3) ∨ (κ2 ∧ κ3). (5) κ1 ∨ (κ2 ∧ κ3) = κ1 ∨ (κ3 ∧ κ2). (6) The operation ∨ is associative in V i.e., κ1 ∨ (κ2 ∨ κ3) = (κ1 ∨ κ2) ∨ κ3. (7) The set Vµ1 = {κ1 ∈ V | µ1 ≤ κ1} = {µ1 ∨ κ1 | κ1 ∈ V } is a distributive lattice under induced operations ∨ and ∧ with µ1 as its least element. (8) κ4 ∨ {κ1 ∧ (κ2 ∧ κ3)} = κ4 ∨ {(κ1 ∧ κ2) ∧ κ3}. (9) κ1 ∨ (κ2 ∨ κ3) = κ1 ∨ (κ3 ∨ κ2). (10) κ1 ∨ κ2 = 1 if and only if κ2 ∨ κ1 = 1. (11) κ1 ∧ κ2 = κ2 ∧ κ1 whenever κ1 ∨ κ2 = 1. Theorem 1. An algebra (V,∨,∧, 1) of type (2, 2, 0) is a PDL if and only if it satisfies the following: (LD∨) κ1 ∨ (κ2 ∧ κ3) = (κ1 ∨ κ2) ∧ (κ1 ∨ κ3) (RD∨) (κ1 ∧ κ2) ∨ κ3 = (κ1 ∨ κ3) ∧ (κ2 ∨ κ3) (RD∧) (κ1 ∨ κ2) ∧ κ3 = (κ1 ∧ κ3) ∨ (κ2 ∧ κ3) (L1) (κ1 ∨ κ2) ∧ κ2 = κ2 (L3) κ1 ∨ (κ1 ∧ κ2) = κ1 (I1) κ1 ∨ 1 = 1 (I2) 1 ∧ κ1 = κ1. for all κ1, κ2, κ3 ∈ V . Definition 2. A Paradistributive Latticoid (V,∨,∧, 1) is said to be associative if it satisfies the following condition κ1 ∧ (κ2 ∧ κ3) = (κ1 ∧ κ2) ∧ κ3 for all κ1, κ2, κ3 ∈ V. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1133 Let V be a PDL. Then, an element µ1 ∈ V is said to be a minimal element if for any u ∈ V , u ≤ µ1 ⇒ u = µ1. Lemma 2. Let V be a PDL. Then, for any µ1 ∈ V, the following are equivalent: (1). µ1 is minimal (2). κ1 ∧ µ1 = µ1 for all κ1 ∈ V (3). κ1 ∨ µ1 = κ1 for all κ1 ∈ V . Definition 3. A non-empty subset F of a PDL V is said to be a filter if it satisfies the following: κ1, κ2 ∈ F ⇒ κ1 ∧ κ2 ∈ F. κ1 ∈ F, µ1 ∈ V ⇒ µ1 ∨ κ1 ∈ F. Theorem 2. Let S be a non-empty subset of V . Then [S) = {κ1 ∨ ( n ∧ i=1 si) | si ∈ S, κ1 ∈ V, 1 ≤ i ≤ n and n is a positive integer } is the smallest filter of V containing S. Lemma 3. Let V be a PDL and F be a filter of V . Then for any κ1, κ2 ∈ V , we have the following: (1) [κ1) = {ρ ∨ κ1 | ρ ∈ V }. (2) κ1 ∈ [κ2) if and only if κ1 = κ1 ∨ κ2 for all κ1, κ2 ∈ V . (3) κ1 ∨ κ2 ∈ F if and only if κ2 ∨ κ1 ∈ F . (4) [κ1 ∨ κ2) = [κ2 ∨ κ1). (5) [κ1 ∧ κ2) = [κ2 ∧ κ1) = [κ1) ∨ [κ2). Theorem 3. The collection F (L) of all filters of a PDL V forms a distributive lattice under set inclusion, in which, the glb and lub of any F and G are given respectively by F ∧G = F ∩G and F ∨G = {κ1 ∧ κ2 | κ1 ∈ F and κ2 ∈ G}. Definition 4. By a homomorphism of a PDL (V,∨,∧, 1) into a PDL (V ′,∨′,∧′, 1′), we mean, a mapping f : V → V ′ satisfying the following: (1) f(µ1 ∨ µ2) = f(µ1) ∨′ f(µ2) (2) f(µ1 ∧ µ2) = f(µ1) ∧′ f(µ2) (3) f(1) = f(1′). 3. Parapseudo-Complementation on Paradistributive Latticoids In this section, we define a parapseudo-complementation on a PDL and present some fundamental findings which helps in verification of the axioms independency. Definition 5. Let (V,∨,∧, 1) be a Paradistributive Latticoid (PDL) and consider a unary operation denoted as ρ 7→ ρ♦ on V . This operation is called a parapseudo-complementation on V if it satisfies the following conditions: (PPC1) If ρ ∨ ϱ = 1, then ρ ∨ ϱ♦ = ρ. (PPC2) ρ ∨ ρ♦ = 1. (PPC3) (ρ ∧ ϱ)♦ = ρ♦ ∨ ϱ♦. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1134 If there is no ambiguity about the parapseudo-complementation on a PDL V , we can say that V is a parapseudo-complemented PDL (PPDL). In the case of a distributive lattice with one, PPC3 becomes a consequence of PPC1 and PPC2. However, in the case of PDLs, PPC1, PPC2, and PPC3 are independent. Now, we provide examples to demonstrate the independence of these axioms. Example 2. Consider a PDL V with at least two elements. Let’s define the unary oper- ation ρ♦ = 1 for all ρ ∈ V . We will show that V satisfies (PPC2) and (PPC3) but fails to satisfy (PPC1). (PPC2): For any ρ ∈ V , we have ρ ∨ ρ♦ = ρ ∨ 1 = 1. Hence, (PPC2) is satisfied. (PPC3): Let ρ, ϱ ∈ V . We have (ρ ∧ ϱ)♦ = 1 and ρ♦ ∨ ϱ♦ = 1 ∨ 1 = 1. Therefore, (PPC3) is satisfied. Now, we examine (PPC1). Suppose there exists ϱ ∈ V such that ϱ ̸= 1. We have ϱ ∨ 1 = 1. However, ϱ ∨ 1♦ = ϱ ∨ 1 = 1 ̸= ϱ. Therefore, (PPC1) is not satisfied when ϱ is not equal to 1. In conclusion, the PDL V with the unary operation ρ♦ = 1 satisfies (PPC2) and (PPC3) but fails to satisfy (PPC1) when V has at least two elements. Example 3. Let V be a bounded distributive lattice with bounds 0 ̸= 1. Define ρ♦ = 0 for all ρ ∈ V . We will show that V satisfies (PPC1) and (PPC3) but fails to satisfy (PPC2). (PPC1): Suppose ρ ∨ ϱ = 1, where ρ, ϱ ∈ V . We have ρ ∨ ϱ♦ = ρ ∨ 0 = ρ. Therefore, (PPC1) is satisfied. (PPC3): For any ρ, ϱ ∈ V , we have (ρ ∧ ϱ)♦ = 0 and ρ♦ ∨ ϱ♦ = 0 ∨ 0 = 0. Thus, (PPC3) is satisfied. Now we examine (PPC2). Suppose 0 ∈ V . We have 0∨ 0♦ = 0∨ 0 = 0, but we require 0 ∨ 0♦ = 1. Therefore, (PPC2) is not satisfied in this case. In conclusion, the bounded distributive lattice V with the unary operation ρ♦ = 0 satisfies (PPC1) and (PPC3) but fails to satisfy (PPC2) when V contains 0 as an element. Example 4. Let V be a disconnected PDL with atleast two elements other than 1. Then (V 3,∨,∧, 1) is a PDL, where ∨,∧ are defined co-ordinate wise. Now, for any ρ ∈ V 3, we write |ρ| for the number of non-units in ρ. Define ♦ on V 3 as follows: For any ρ ∈ V 3, define ρ♦ = (ρ♦1 , ρ ♦ 2 , ρ ♦ 3 ) where, for i = 1, 2, 3 ρ♦i =  1 ρi ̸= 1 0 ρi = 1 |ρ| = 2 2 ρi = 1 |ρ| = 1, |ρ| > 2 and 1♦ = (2, 2, 2). Then (V 3,∨,∧, 1) is a PPDL which satisfies (PPC1) and (PPC2) but fails to satisfy (PPC3). For, if ρ = (0, 1, 1) and ϱ = (1, 0, 1), then ρ♦ = (1, 2, 2) and ϱ♦ = (2, 1, 2) and ρ ∧ ϱ = (0, 0, 1). Hence (ρ ∧ ϱ)♦ = (1, 1, 0) and ρ♦ ∨ ϱ♦ = (1, 1, 2). Therefore, (ρ ∧ ϱ)♦ ̸= ρ♦ ∨ ϱ♦. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1135 Lemma 4. Let (V,+, ·, 0, 1) be a commutative regular ring with unity and let ρ0 be the unique idempotent element in V such that ρV = ρ0V . Now, for any ρ, ϱ ∈ V , define (1) ρ ∨ ϱ = ϱ0ρ (2) ρ ∧ ϱ = ρ+ ϱ− ϱ0ρ (3) ρ♦ = 1− ρ0. Then (V,∨,∧, 0) is a PDL in which 1 is a minimal element and ♦ is a parapseudo- complementation on V . Proof. It is clear that (V,∨,∧, 0) is a PDL. Note that, for any ρ, ϱ ∈ V , (ρϱ)0 = ρ0ϱ0 and (ρ+ ϱ− ρ0ϱ)0 = ρ0 + ϱ0 − ρ0ϱ0. Also, 00 = 0 and 10 = 1. Now, we prove that ♦ is a parapseudo-complementation on V . Let ρ, ϱ ∈ V and ρ ∨ ϱ = 0. Then ϱ0ρ = 0 and ρ ∨ ϱ♦ = (ϱ♦)0ρ = ϱ♦ρ = (1− ϱ0)ρ = ρ Also, ρ ∨ ρ♦ = ρ ∨ (1− ρ0) = (1− ρ0)0ρ = (1− ρ0)ρ = ρ− ρ = 0. Let ρ, ϱ ∈ V . Then, ρ♦ ∨ ϱ♦ = (1− ρ0) ∨ (1− ϱ0) = (1− ϱ0)0(1− ρ0) = (10 − ϱ0)(1− ρ0) = (1− ϱ0)(1− ρ0) = 1− ρ0 − ϱ0 + ρ0ϱ0 = 1− (ρ0 + ϱ0 − ρ0ϱ0) = 1− (ρ+ ϱ− ϱ0ρ)0 = 1− (ρ ∧ ϱ)0 = (ρ ∧ ϱ)♦ Therefore, ♦ is a parapseudo-complementation on V . Example 5. Let (V,∨,∧, 1) be a disconnected PDL. Fix ρ1 ̸= 1 ∈ V and define ♦ on V as follows: µ♦ 1 = { 1 µ1 ̸= 1 ρ1 µ1 = 1 Then ♦ is a parapseudo-complementation on V . In the case of a distributive lattice the dual pseudo-complementation, if exists, is unique. But, in a PDL there can be several parapseudo-complementations. For, in Exam- ple 5, we get one parapseudo-complementation on V corresponding to each ρ1(̸= 1) ∈ V . Theorem 4. Every finite PDL is parapseudo-complemented. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1136 Proof. Let V be a finite PDL. Then V has a minimal element, say m. Now, we prove that V is parapseudo-complemented. For this, define ♢ on V by ρ♢ = (m ∨ ρ)♦, where (m ∨ ρ)♦ is the dual pseudo-complement of m ∨ ρ in the finite distributive lattice [m, 1] and ρ ∈ V . We prove ♢ is parapseudo-complementation on V . Let ρ, ϱ ∈ V . Then ρ♢ ∨ ρ = ρ♢ ∨ (ρ ∨ m) = (m ∨ ρ)♦ ∨ (m ∨ ρ) = 1. Suppose ϱ ∨ ρ = 1. Then (m ∨ ρ) ∨ (m ∨ ϱ) = 1 and m ∨ ϱ ∈ [m, 1]. Hence (m ∨ ρ)♦ ≤ (m ∨ ϱ). Thus ρ♢ ≤ (m ∨ ϱ). Now, ϱ ≤ ϱ ∨ ρ♢ ≤ ϱ ∨ m ∨ ϱ = ϱ. Therefore, ϱ ∨ ρ♢ = ϱ. Let ρ, ϱ ∈ V . Then (ρ ∧ ϱ)♢ = (m ∨ (ρ ∧ ϱ))♦ = ((m ∨ ρ) ∧ (m ∨ ϱ))♦ = (m ∨ ρ)♦ ∨ (m ∨ ϱ)♦ = ρ♢ ∨ ϱ♢. Let (V,∨,∧, 1) be a PDL. By an interval in V , we mean the set [ρ, ϱ] = {µ1 ∈ V |ρ ≤ µ1 ≤ ϱ} for some ρ, ϱ ∈ V such that ρ ≤ ϱ. Clearly, [ρ, ϱ] is closed under ∨,∧. Since [ρ, ϱ] is a PDL with ρ as its zero element and ϱ as its greatest element, every interval [ρ, ϱ] is a bounded distributive lattice. Definition 6. A PDL (V,∨,∧, 1) is said to be relatively complemented if every interval [ρ, ϱ], ρ ≤ ϱ in V is a complemented lattice. Theorem 5. Let (V,∨,∧, 1) be a PDL with 1. Then the following are equivalent: (1). V is relatively complemented. (2). V is sectionally complemented, i.e the interval [ρ, 1], ρ ∈ V is a complemented lattice. (3). Given ρ, ϱ ∈ V , there exists a unique µ1 ∈ V such that µ1 ∨ ρ = 1 and µ1 ∧ ρ = ϱ∧ ρ. Proof. (1) ⇒ (2) is clear. (2) ⇒ (3) : Assume (2) and let ρ, ϱ ∈ V . So that the interval [ϱ ∧ ρ, 1] is complemented and ρ ∈ [ϱ ∧ ρ, 1]. If µ1 is the complement of ρ in [ϱ ∧ ρ, 1], then µ1 ∧ ρ = ϱ ∧ ρ and µ1 ∨ ρ = 1. Since any µ2 ∈ V satisfies µ2 ∧ ρ = ϱ ∧ ρ and µ2 ∨ ρ = 1 belongs to [ϱ ∧ ρ, 1]. Therefore, [ϱ ∧ ρ, 1] is a boolean algebra and hence the uniqueness of µ1 follows. (3) ⇒ (1) : Assume (3). Let ρ, ϱ ∈ V such that ϱ ≤ ρ and let µ1 ∈ [ϱ, ρ]. Then by (3), there exists µ2 ∈ V such that µ1 ∨ µ2 = 1 , µ2 ∧ µ1 = ϱ ∧ µ1 = ϱ. It is clear that ϱ = µ2 ∧ µ1 = µ1 ∧ µ2 ≤ µ2. Now, we prove that the element µ2 ∧ ρ ∈ [ϱ, ρ] and µ2 ∧ ρ is complement of µ1 in [ϱ, ρ]. Clearly µ2 ∧ ρ ≤ ρ. Now, ϱ ∨ (µ2 ∧ ρ) = (ϱ ∨ µ2) ∧ (ϱ ∨ ρ) = µ2 ∧ ρ. Hence µ2 ∧ ρ ∈ [ϱ, ρ]. Now, µ1 ∨ (µ2 ∧ ρ) = (µ1 ∨ µ2) ∧ (µ1 ∨ ρ) = 1 ∧ (µ1 ∨ ρ) = ρ and ϱ = µ2 ∧ µ1 = [µ2 ∨ (µ2 ∧ ρ)] ∧ µ1. = (µ2 ∧ µ1) ∨ [(µ2 ∧ ρ) ∧ µ1]. = (µ2 ∧ µ1) ∨ [µ1 ∧ (µ2 ∧ ρ)]. = [(µ2 ∧ µ1) ∨ µ1] ∧ [(µ2 ∧ µ1) ∨ (µ2 ∧ ρ)]. = µ1 ∧ [ϱ ∨ (µ2 ∧ ρ)]. = µ1 ∧ [(ϱ ∨ µ2) ∧ (ϱ ∨ ρ)]. = µ1 ∧ (µ2 ∧ ρ). Therefore, µ2 ∧ ρ is the complement of µ1 in [ϱ, ρ]. Hence V is relatively complemented. Note that every relatively complemented PDL is an associative PDL. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1137 Theorem 6. Let V be a relatively complemented PDL with a minimal element m1. Then V is parapseudo-complemented PDL. Proof. Let V be a relatively complemented PDL with a minimal element m1. For any ρ ∈ V , let ρ♦ be the complement of ρ ∈ [m1 ∧ ρ, 1]. Now, we prove that ♦ is parapseudo- complementation on V . Clearly, for any ρ ∈ V , we have ρ∨ρ♦ = 1. Let ϱ ∈ V be such that ϱ∨ρ = 1. Then, ϱ∨ρ♦ = ϱ∨ (ρ♦∧ρ) = ϱ∨ (m1∧ρ) = ϱ∨m1 = ϱ. Now, we prove that, for any ρ, ϱ ∈ V , (ρ∧ ϱ)♦ = ρ♦ ∨ ϱ♦. Now, (ρ♦ ∨ ϱ♦)∨ (ρ∧ ϱ) = (ρ♦ ∨ ϱ♦ ∨ ρ)∧ (ρ♦ ∨ ϱ♦ ∨ ϱ) = (ρ♦ ∨ ρ ∨ ϱ♦) ∧ 1 = 1 ∧ 1 = 1. Then, (ρ♦ ∨ ϱ♦) ∧ (ρ ∧ ϱ) = (ρ♦ ∧ ρ ∧ ϱ) ∨ (ϱ♦ ∧ ρ ∧ ϱ) = (m1∧ρ∧ϱ)∨(ϱ♦∧ϱ∧ρ) = (m1∧ρ∧ϱ)∨(m1∧ϱ∧ρ) = (m1∧ρ∧ϱ)∨(m1∧ρ∧ϱ) = (m1∧ρ∧ϱ). 4. Properties We present here some elementary properties of parapseudo-complemented PDL and further prove some essential conditions for a PDL with a minimal element to be parapseudo- complemented. The following lemma can be proved easily. Lemma 5. Let V be a parapseudo-complemented PDL. Then, for any ρ, ϱ ∈ V , we have the following: (1). 1♦ is a minimal element. (2). If ρ is a minimal element, then ρ♦ = 1. (3). 1♦ ♦ = 1. (4). ρ♦ ∨ ρ = 1. (5). ρ ∨ ρ♦ ♦ = ρ. (6). ρ♦ = ρ♦ ♦♦ . (7). ρ♦ = 1 ⇔ ρ♦ ♦ is minimal element. (8). 1♦ ≤ ρ♦. (9). ρ♦ ∨ ϱ♦ = ϱ♦ ∨ ρ♦. (10). ρ ≤ ϱ ⇒ ϱ♦ ≤ ρ♦. (11). (ρ ∨ ϱ)♦ ≤ ϱ♦, (ρ ∨ ϱ)♦ ≤ ρ♦. (12). ρ♦ ≤ ϱ♦ ⇔ ϱ♦ ♦ ≤ ρ♦ ♦ . (13). ρ = 1 ⇔ ρ♦ ♦ = 1. Lemma 6. Let V be a PDL with two minimal elements m1 and m2. Then the bounded distributive lattices [m1, 1] and [m2, 1] are isomorphic. Proof. Let V be a PDL with two minimal elements, m1 and m2. Define f : [m1, 1] → [m2, 1] by f(ρ) = m2 ∨ ρ. Now, we prove that f is an isomorphism. Clearly f is well defined. Let ρ, ϱ ∈ [m1, 1] and f(ρ) = f(ϱ). Then m2 ∨ ρ = m2 ∨ ϱ. Now ρ = m1 ∨ ρ = m1 ∨ m2 ∨ ρ = m1 ∨ m2 ∨ ϱ = m1 ∨ ϱ = ϱ. Therefore, f is one-one. Let t ∈ [m2, 1]. Then m1 ∨ t ∈ [m1, 1] and f(m1 ∨ t) = m2 ∨ m1 ∨ t = m2 ∨ t = t. Hence, f is onto. Let ρ, ϱ ∈ [m1, 1]. Then f(ρ ∨ ϱ) = m2 ∨ ρ ∨ ϱ = (m2 ∨ ρ) ∨ (m2 ∨ ϱ) = f(ρ) ∨ f(ϱ) and f(ρ ∧ ϱ) = m2 ∨ (ρ ∧ ϱ) = (m2 ∨ ρ) ∧ (m2 ∨ ϱ) = f(ρ) ∧ f(ϱ), which implies f satisfies homomorphism property. Also, f(1) = m2 ∨ 1 = 1. Therefore, f is an isomorphism. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1138 Theorem 7. Let V be a PDL with a minimal element, m. Then the following are equiv- alent: (1). V is a parapseudo-complemented PDL. (2). [m, 1] is a dual pseudo-complemented lattice. (3). [m1, 1] is a dual pseudo-complemented lattice for all minimal elements m1 in V . Proof. (1) ⇒ (2) : Let ♢ be a parapseudo-complementation on V . We know that [m, 1] is a bounded distributive lattice. Now define ♦ on [m, 1] by ρ♦ = m∨ ρ♢ for all ρ ∈ [m, 1]. Then ρ♦ ∈ [m, 1] and ρ ∨ ρ♦ = ρ ∨ (m ∨ ρ♢) = ρ ∨ ρ♢ ∨m = 1 ∨m = 1. Let ϱ ∈ [m, 1] and ρ∨ ϱ = 1. Then ϱ∨ ρ = 1 and hence ϱ∨ ρ♢ = ϱ. Now ρ♦ ∨ ϱ = m∨ ρ♢ ∨ ϱ = m∨ ϱ∨ ρ♢ = m ∨ ϱ = ϱ. Therefore, ρ♦ ≤ ϱ. Hence, ♦ is dual pseudo-complementation on [m, 1]. (2) ⇒ (3) : Suppose [m, 1] is a dual pseudo-complemented lattice. Then [m1, 1] is a dual pseudo-complemented lattice for all minimal elements m1 in V . (3) ⇒ (1) : Suppose [m1, 1] is a dual pseudo-complemented lattice for all minimal elements m1 in V . For any ρ ∈ V , we have m∧ ρ is a minimal element in V and [m∧ ρ, 1] is a dual pseudo-complemented lattice. Let ρ♦ be the dual pseudo-complement of ρ in [m ∧ ρ, 1]. We prove that ρ ↣ ρ♦ is a parapseudo-complementation on V . Clearly, ρ ∨ ρ♦ = 1. Let ϱ ∈ V and ϱ∨ ρ = 1. Put τ = (m∧ ρ)∨ ϱ = (m∨ ϱ)∧ (ρ∨ ϱ) = (m∨ ϱ)∧ 1 = m∨ ϱ. Then τ ∈ [m∧ ρ, 1] and ρ∨ τ = ρ∨m∨ ϱ = ρ∨ ϱ = 1. So that ρ♦ ≤ τ. Now, τ = ρ♦ ∨ τ implies that m ∨ ϱ = ρ♦ ∨m ∨ ϱ and hence ϱ ∨m ∨ ϱ = ϱ ∨ ρ♦ ∨ ϱ. Thus, we get ϱ = ϱ ∨ ρ♦. Finally, let ρ, ϱ ∈ V . We have ρ ∈ [m ∧ ρ, 1] and ϱ ∈ [m ∧ ϱ, 1]. Now, (m ∧ (ρ ∧ ϱ)) ∨ (ρ♦ ∨ ϱ♦) = (m ∨ ρ♦ ∨ ϱ♦) ∧ ((ρ ∧ ϱ) ∨ ρ♦ ∨ ϱ♦). = (m ∨ ρ♦ ∨ ϱ♦) ∧ (ρ ∨ ρ♦ ∨ ϱ♦) ∧ (ϱ ∨ ρ♦ ∨ ϱ♦) = (m ∨ ρ♦ ∨ ϱ♦). = ρ♦ ∨ ϱ♦ ( since ρ♦ ∈ [m ∧ ρ, 1]) Therefore, ρ♦ ∨ ϱ♦ ∈ [m ∧ (ρ ∧ ϱ), 1]. Now, (ρ ∧ ϱ) ∨ (ρ♦ ∨ ϱ♦) = (ρ ∨ ρ♦ ∨ ϱ♦) ∧ (ϱ ∨ ρ♦ ∨ ϱ♦) = 1. Let τ ∈ [m ∧ (ρ ∧ ϱ), 1] and (ρ ∧ ϱ) ∨ τ = 1. Then, (ρ ∨ τ) ∧ (ϱ ∨ τ) = 1 which implies that ρ ∨ τ = 1 and ϱ ∨ τ = 1. Also, (m∧ ρ)∨ τ ∈ [m∧ ρ, 1] and ρ∨ ((m∧ ρ)∨ τ) = ρ∨ (τ ∨ (m∧ ρ)) = 1. Hence, we get ρ♦ ≤ (m∧ρ)∨τ . So that, (m∧ρ)∨τ = ρ♦∨ (m∧ρ)∨τ = [(ρ♦∨m)∧ (ρ♦∨ρ)]∨τ = ρ♦∨τ. Therefore, m ∨ τ = ρ♦ ∨ τ. Now, [m ∧ (ρ ∧ ϱ)] ≤ τ implies τ = [m ∧ (ρ ∧ ϱ)] ∨ τ = (m ∨ τ) ∧ [(ρ ∨ τ) ∧ (ϱ ∨ τ)] = m ∨ τ. Therefore, ρ♦ ∨ τ = τ implies ρ♦ ≤ τ . Similarly, we get ϱ♦ ≤ τ . Hence ρ♦ ∨ ϱ♦ ≤ τ . Thus, we get that (ρ ∧ ϱ)♦ = ρ♦ ∨ ϱ♦. Therefore, ♦ is a parapseudo-complementation on V . Lemma 7. Let V be a PDL and A ⊆ V . Then the set A• = {t ∈ V | t ∨ ρ = 1 for all ρ ∈ A} is a filter of V . R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1139 Proof. Let t1, t2 ∈ A•. Then t1 ∨ ρ = 1, t2 ∨ ρ = 1 for all ρ ∈ V . Hence (t1 ∧ t2) ∨ ρ = (t1 ∨ ρ)∧ (t2 ∨ ρ) = 1∧ 1 = 1. Therefore, t1 ∧ t2 ∈ A•. Now, let t1 ∈ A• and κ1 ∈ V . Then κ1 ∨ t1 ∨ ρ = κ1 ∨ 1 = 1. Hence, we get κ1 ∨ t1 ∈ A•. Thus, A• is a filter of V . The filter A• is called the annihilator filter corresponding to A. If A = {ρ}, we write A• = [ρ]•. Lemma 8. Let V be a PDL. Then for any ρ, ϱ ∈ V , [ρ ∧ ϱ]• = [ρ]• ∩ [ϱ]•. Proof. Let κ1 ∈ V . Then, κ1 ∈ [ρ∧ ϱ]• ⇔ κ1 ∨ (ρ∧ ϱ) = 1 ⇔ (κ1 ∨ ρ)∧ (κ1 ∨ ϱ) = 1 ⇔ κ1 ∨ ρ = 1 and κ1 ∨ ϱ = 1 ⇔ κ1 ∈ [ρ]• and κ1 ∈ [ϱ]• ⇔ κ1 ∈ [ρ]• ∩ [ϱ]•. Lemma 9. Let V be a PDL and ρ ∈ V . Then [ρ) = V if and only if ρ is a minimal element. Proof. Suppose [ρ) = V . Then, for any κ1 ∈ V , we have κ1 ∈ [ρ) and hence κ1∨ρ = κ1. Therefore, ρ is a minimal element. Conversely, suppose that ρ is a minimal element. We have [ρ) ⊆ V . Let κ1 ∈ V . Then κ1 ∨ ρ = κ1. Therefore κ1 ∈ [ρ). Hence [ρ) = V . Now, we prove the following theorem which characterize parapseudo-complementation on PDL. Theorem 8. Let V be a PDL. Then V is a parapseudo-complemented PDL if and only if for any ρ ∈ V , the annihilator filter [ρ]• is a principal filter. Proof. Let ρ ∈ V be such that [ρ]• = [κ1) for some κ1 ∈ V . Since 1 ∈ V , we have V = [1]• = [m) for some m ∈ V . Hence by Lemma 9, m is a minimal element in V . Define ρ♢ = m ∨ κ1. Now, we prove that ♢ is a parapseudo-complementation on V . Let ρ ∈ V and suppose [ρ]• = [κ1) = [κ2) for some κ1, κ2 ∈ V . Then κ1 = κ1 ∨ κ2 and κ2 = κ2 ∨ κ1. Therefore, m∨κ1 = m∨κ1∨κ2 = m∨κ2∨κ1 = m∨κ2 which implies ♢ is well-defined. Let ρ ∈ V . Then ρ∨ρ♢ = ρ∨m∨κ1 = ρ∨κ1 = 1. Let ϱ ∈ V and ϱ∨ρ = 1. Then ϱ ∈ [ρ]• = [κ1). Therefore, ϱ = ϱ ∨ κ1 = ϱ ∨ m ∨ κ1 = ϱ ∨ ρ♢. Finally, let ρ, ϱ ∈ V and [ρ]• = [κ1), [ϱ]• = [κ2) for some κ1, κ2 ∈ V . Then, [ρ ∧ ϱ]• = [ρ]• ∩ [ϱ]• = [κ1) ∩ [κ2) = [κ1 ∨ κ2). Therefore (ρ∧ϱ)♢ = m∨ (κ1∨κ2) = (m∨κ1)∨ (m∨κ2) = ρ♢∨ϱ♢. Thus ♢ is parapseudo- complementation on V . Conversely, if V is parapseudo-complemented PDL, then for any ρ ∈ V , we have [ρ]• = [ρ♦). Hence, every annihilator filter is a principal filter. Theorem 9. Let V be a PDL with a minimal element m. Then V is parapseudo- complemented PDL if and only if the set PF(V ) of all principal filters of V is a pseudo- complemented lattice. Proof. Suppose V is a parapseudo-complemented PDL. Then the set PF(V ) forms a distributive lattice. Let [ρ) ∈ PF (V). Define [ρ)♢ = [ρ♦) where ρ♦ is the parapseudo- complement of ρ ∈ V . We prove that ♢ is a pseudo-complementation on PF(V ). Now, R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1140 [ρ) ∩ [ρ)♢ = [ρ) ∩ [ρ♦) = [ρ ∨ ρ♦) = [1). Let τ ∈ V and [ρ) ∩ [τ) = [1). Then [ρ ∨ τ) = [1) implies ρ ∨ τ = 1. Hence τ ∨ ρ♦ = τ . Therefore, [τ) ∩ [ρ♦) = [τ ∨ ρ♦) = [τ) so that [τ) ⊆ [ρ♦) = [ρ)♢. Thus [ρ)♢ is the pseudo-complement of [ρ) in PF(V ). Conversely, suppose PF(V ) is a pseudo-complemented lattice. Let ρ ∈ V . Then [ρ) ∈ PF (V). Write [ρ)♦ = [ρ1) the pseudo-complement of [ρ) ∈ PF (V). Now, define ρ♢ = m ∨ ρ1. Then we prove that ♢ is a parapseudo-complementation on V . First we observe that ♢ is well defined. Suppose [ρ)♦ = [ρ1) = [ρ2). Then m ∨ ρ1 = m ∨ ρ1 ∨ ρ2 = m ∨ ρ2 ∨ ρ1 = m ∨ ρ2. Hence ♢ is well defined. Now, ρ ∨ ρ♢ = ρ ∨m ∨ ρ1 = ρ ∨ ρ1 = 1, since [ρ ∨ ρ1) = [ρ) ∩ [ρ1) = [ρ) ∩ [ρ)♦ = [1). Let ϱ ∈ V and ϱ ∨ ρ = 1. Then, [ρ) ∩ [ϱ) = [1) and hence [ϱ) ∩ [ρ)♦ = [ϱ). Therefore [ϱ) ∩ [ρ1) = [ϱ) which implies that [ϱ) ⊆ [ρ1). Hence ϱ = ϱ ∨ ρ1 = ϱ ∨ ρ1 ∨ m = ϱ ∨ ρ♢. Finally, let ρ, ϱ ∈ V and suppose [ρ)♦ = [ρ1), [ϱ) ♦ = [ϱ1) for some ρ1, ϱ1 ∈ V . Now, [ρ∧ϱ)♦ = [ρ)♦∩ [ϱ)♦ = [ρ1)∩ [ϱ1) = [ρ1∨ϱ1). Hence, by definition, (ρ∧ϱ)♢ = m∨ρ1∨ϱ1 = m ∨ ρ1 ∨m ∨ ϱ1 = ρ♢ ∨ ϱ♢. Thus ♢ is a parapseudo-complementation on V . In 1949, P.Ribenboim[10] had first observed that the class of pseudo-complemented distributive lattices is equational. Now, we prove that the parapseudo-complementation on paradistributive latticoids is also equationally definable. We give certain equivalent sets of identities which characterize the parapseudo-complementation on V . For this, first we need the following lemmas. As there are no hidden difficulties to prove the following two lemmas, we omit their proofs. Lemma 10. Let V be a parapseudo-complemented PDL. Then for any ρ, ϱ ∈ V , the following are equivalent: (1). ρ ∨ ϱ = 1. (2). ρ ∨ ϱ♦ ♦ = 1. (3). ρ♦ ♦ ∨ ϱ♦ ♦ = 1. (4). ρ ∨ ϱ♦ ♦ = 1. Lemma 11. Let V be a parapseudo-complemented PDL. Then for any ρ, ϱ ∈ V , the following hold: (1). (ρ ∨ ϱ)♦ ♦ = ρ♦ ♦ ∨ ϱ♦ ♦ . (2). (ρ ∨ ϱ)♦ = (ϱ ∨ ρ)♦. (3). (ρ ∧ ϱ)♦ = (ϱ ∧ ρ)♦. Now, we prove that the parapseudo-complementation PDL is equationally definable. Lemma 12. A unary operation ♦ on PDL V is a parapseudo-complementation on V if and only if it satisfies the following equations: (1). ρ ∨ ρ♦ = 1. (2). ρ ∧ ρ♦ ♦ = ρ♦ ♦ . (3). (ρ ∧ ϱ)♦ = ρ♦ ∨ ϱ♦. (4). (ρ ∨ ϱ)♦ ♦ = ρ♦ ♦ ∨ ϱ♦ ♦ . (5). ρ ∨ 1♦ = ρ. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1141 Proof. Let V be a PDL and ♦ be a unary operation on V satisfying the given conditions. We prove that ♦ is a parapseudo-complementation on V . Let ρ, ϱ ∈ V and ϱ∨ρ = 1. Then, ϱ = ϱ ∨ ϱ♦ ♦ ( by (2)) = ϱ ∨ 1♦ ∨ ϱ♦ ♦ ( by (5)) = ϱ ∨ (ρ♦ ∨ ρ♦ ♦ )♦ ∨ ϱ♦ ♦ ( by (1)) = ϱ ∨ (ρ ∧ ρ♦)♦ ♦ ∨ ϱ♦ ♦ ( by (3)) = ϱ ∨ ((ρ ∧ ρ♦) ∨ ϱ)♦ ♦ ( by (4)) = ϱ ∨ ((ρ ∨ ϱ) ∧ (ρ♦ ∨ ϱ))♦ ♦ = ϱ ∨ (ρ♦ ∨ ϱ)♦ ♦ = ϱ ∨ ρ♦ ♦♦ ∨ ϱ♦ ♦ ( by (4)) = ϱ ∨ ρ♦ ∨ ϱ♦ ♦ ( by (2)) = ϱ ∨ ϱ♦ ♦ ∨ ρ♦ ( by (3)) = ϱ ∨ ρ♦ ( by (2)) Therefore, it follows from (1) and (3) that ♦ is a parapseudo-complementation on V . Converse follows from Lemma 10 and Lemma 11. Lemma 13. A unary operation ♦ on PDL V is a parapseudo-complementation on V if and only if it satisfies the following equations: (1). ϱ ∨ ρ♦ = ϱ ∨ (ρ ∨ ϱ)♦. (2). ρ ∨ 1♦ = ρ. (3). 1♦ ♦ = 1. (4). (ρ ∧ ϱ)♦ = ρ♦ ∨ ϱ♦. Proof. Let V be a PDL and ♦ be a unary operation on V satisfying the given equations. We prove ♦ is a parapseudo-complementation on V . Let ρ, ϱ ∈ V and ϱ ∨ ρ = 1. Then ϱ ∨ ρ♦ = ϱ ∨ (ρ ∨ ϱ)♦ = ϱ ∨ 1♦ = ϱ. Now ρ ∨ ρ♦ = ρ ∨ (ρ ∨ 1♦)♦ = ρ ∨ (1♦ ∨ ρ)♦ = ρ ∨ 1♦ ♦ = ρ ∨ 1 = 1 shows that ♦ is a parapseudo-complementation on V . Conversely, assume that ♦ is a parapseudo-complementation on V . Then, by Lemma 5 and by Definition 5 we have (2), (3), (4). So, it is enough if we prove (1). For this, let ρ, ϱ ∈ V . Then ρ ∨ ϱ ∨ (ρ ∨ ϱ)♦ = 1 ⇒ ϱ ∨ (ρ ∨ ϱ)♦ ∨ ρ♦ = ϱ ∨ (ρ ∨ ϱ)♦ ⇒ ϱ ∨ ρ♦ ∨ (ρ ∨ ϱ)♦ = ϱ ∨ (ρ ∨ ϱ)♦ ⇒ ϱ ∨ ρ♦ = ϱ ∨ (ρ ∨ ϱ)♦ R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1142 5. One to One Correspondence In this section, we prove that, if ♦ is a parapseudo-complementation on V , then the set V ♦ = {ρ♦ | ρ ∈ V } is a Boolean algebra. Furthermore, there exists a one-to-one cor- respondence between the set of all minimal elements of V and the set of all parapseudo- complementations on V . Finally, it is worth noting that the Boolean algebra V ♦ is inde- pendent of the specific choice of parapseudo-complementation ♦. Theorem 10. Let V be a PDL with a parapseudo-complementation ♦. For any ρ♦, ϱ♦ ∈ V ♦, define ρ♦ ≤ ϱ♦ if and only if ρ♦ ∧ ϱ♦ = ρ♦. Then (V ♦,≤) is a Boolean algebra. Proof. Let V be a PDL with a parapseudo-complementation ♦. Clearly, ≤ is reflexive and anti-symmetric. Now, for ρ♦ ≤ ϱ♦ and ϱ♦ ≤ τ♦, we have ρ♦ ∧ ϱ♦ = ρ♦ , ϱ♦ ∧ τ♦ = ϱ♦. Therefore, ρ♦∨τ♦ = ρ♦∨ϱ♦∨τ♦ = ϱ♦∨τ♦ = τ♦ which implies ρ♦∧τ♦ = ρ♦∧(ρ♦∨τ♦) = ρ♦, hence ≤ is transitive. Therefore, ≤ is a partial ordering on V ♦. Let ρ♦, ϱ♦ ∈ V ♦. Then (ρ ∧ ϱ)♦ = ρ♦ ∨ ϱ♦. Hence ρ♦ ∨ ϱ♦ ∈ V ♦ and we have ρ♦ ∨ ϱ♦ = ϱ♦ ∨ ρ♦. So that, ρ♦ ∨ ϱ♦ is the least upper bound of ρ♦, ϱ♦ ∈ V ♦. We have ρ♦ ♦ ≤ ρ♦ ♦ ∨ ϱ♦ ♦ implies (ρ♦ ♦ ∨ ϱ♦ ♦ )♦ ≤ ρ♦. Similarly, we get (ρ♦ ♦ ∨ ϱ♦ ♦ )♦ ≤ ϱ♦. Therefore, (ρ♦ ♦ ∨ ϱ♦ ♦ )♦ is a lower bound of ρ♦, ϱ♦ ∈ V ♦. Let τ♦ ∈ V ♦ and τ♦ ≤ ρ♦ , τ♦ ≤ ϱ♦. Then ρ♦ ♦ ≤ τ♦ ♦ , ϱ♦ ♦ ≤ τ♦ ♦ . Hence ρ♦ ♦ ∨ϱ♦ ♦ ≤ τ♦ ♦ . Therefore, τ♦ ≤ (ρ♦ ♦ ∨ϱ♦ ♦ )♦. Thus (ρ♦ ♦ ∨ϱ♦ ♦ )♦ is the greatest lower bound of ρ♦, ϱ♦ ∈ V ♦. Hence (V ♦,≤) is a lattice. From now, we represent (ρ♦ ♦ ∨ ϱ♦ ♦ )♦ =(ρ♦∧ϱ♦). Now, by Lemma 5(8), we have 1♦ ≤ ρ♦ for all ρ ∈ V . So that, 1♦ is the least element in V ♦ and since 1♦ ♦ = 1, 1 ∈ V ♦ it is the greatest element in V ♦. Therefore, (V ♦,≤) is a bounded lattice. Finally, we prove that (V ♦,≤) has complement and satisfies distributive property. Let ρ♦ ∈ V ♦. Then ρ♦ ♦ ∈ V ♦ and ρ♦∨ρ♦♦ = 1 and ρ♦∧ρ♦♦ = (ρ♦ ♦∨ρ♦♦♦ )♦ = (ρ♦ ♦∨ρ♦)♦ = 1♦. Hence ρ♦ ♦ is the complement of ρ♦ ∈ V ♦. Now, let ρ♦, ϱ♦, τ♦ ∈ V ♦. Then, (ρ♦∧ϱ♦) ∨ (ρ♦∧τ♦) = (ρ♦ ♦ ∨ ϱ♦ ♦ )♦ ∨ (ρ♦ ♦ ∨ τ♦ ♦ )♦. = [(ρ♦ ♦ ∨ ϱ♦ ♦ ) ∧ (ρ♦ ♦ ∨ τ♦ ♦ )]♦. = [ρ♦ ♦ ∨ (ϱ♦ ♦ ∧ τ♦ ♦ )]♦. = [ρ♦ ♦ ∨ (ϱ♦ ♦ ∧ τ♦ ♦ )]♦ ♦♦ . = [ρ♦ ♦♦♦ ∨ (ϱ♦ ♦ ∧ τ♦ ♦ )♦ ♦ ]♦. = [ρ♦ ♦ ∨ (ϱ♦ ♦♦ ∨ τ♦ ♦♦ )♦]♦. = [ρ♦ ♦ ∨ (ϱ♦ ∨ τ♦)♦]♦. = ρ♦∧(ϱ♦ ∨ τ♦). Thus (V ♦,≤) is a Boolean algebra. Corollary 7. Let V be a parapseudo-complemented PDL with parapseudo-complemenation ♦. Then the map f : V → V ♦ defined by f(ρ) = ρ♦ ♦ is an epimorphism. R. Shukla et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1129-1145 1143 In the following theorem, we establish a one-to-one correspondence between the set of all minimal elements in V and the set of all parapseudo-complemenations on V . First we prove the following lemma. Lemma 14. Let V be a PDL with two parapseudo-complemenations ♦ and ♢. Then, for any ρ, ϱ ∈ V , we have the following: (1). ρ♢ ∨ ρ♦ = ρ♢ and ρ♢ ∧ ρ♦ = ρ♦. (2). ρ♦♢ = ρ♢♢. (3). ρ♦ = ϱ♦ ⇔ ρ♢ = ϱ♢. (4). ρ♦ = 1 ⇔ ρ♢ = 1 ⇔ (ρ ∨ ϱ = 1 ⇒ ϱ = 1). (5). ρ♢ = 1♢ ∨ ρ♦. (6). ρ♦ ∧ ρ♦ ♦ = 1♦ ⇔ ρ♢ ∧ ρ♢ ♢ = 1♢. Proof. Let V be a PDL with two parapseudo-complemenations ♦ and ♢ and ρ, ϱ ∈ V . (1). Since ρ♢ ∨ ρ = 1, we get that ρ♢ ∨ ρ♦ = ρ♢. Hence ρ♢ ∧ ρ♦ = ρ♦. (2). ρ♦♢ = (ρ♢ ∧ ρ♦)♢ = (ρ♦ ∧ ρ♢)♢ = ρ♢♢. (3). Let ρ♦ = ϱ♦. Then ρ♢ = ρ♢♢♢ = ρ♦♢♢ = ϱ♦♢♢ = ϱ♢♢♢ = ϱ♢. (4). Let ρ♦ = 1. Then, we have ρ♢ = ρ♢ ∨ ρ♦ = ρ♢ ∨ 1 = 1. Let ρ♢ = 1 and ρ ∨ ϱ = 1. Then ϱ = ϱ ∨ ρ♢ = ϱ ∨ 1 = 1. Suppose ϱ = 1 whenever ρ ∨ ϱ = 1. So that ρ♦ = 1 since ρ ∨ ρ♦ = 1. (5). We have (1♢ ∨ ρ♦) ∨ ρ♢ = 1♢ ∨ ρ♢ ∨ ρ♦ = ρ♢ ∨ ρ♦ = ρ♢. (6). Let ρ♦∧ρ♦♦ = 1♦. Then ρ♢∧ρ♢♢ = ρ♢∧ρ♦♢ = (1♢∨ρ♦)∧(1♢∨ρ♦♦) = 1♢∨(ρ♦∧ρ♦♦) = 1♢ ∨ 1♦ = 1♢. Let (V,∨,∧, 1) be a PDL with a parapseudo-complementation ♦, and m a minimal element in V . We define a new operation ♦m : V → V as follows: for any ρ ∈ V , we have ρ♦m = m ∨ ρ♦. Then ♦m is also a parapseudo-complementation on V in which 1♦m = m. Theorem 11. Let V be a parapseudo-complemented PDL. Let M be the set of all minimal elements in V and PC(V ) be the set of all parapseudo -complemenations on V . For any m ∈ M , define ♦m : V → V by ρ♦m = m∨ ρ♦ for all ρ ∈ V . Then m ↣ ρ♦m is a bijection of M onto PC(V ). Proof. Let m,n ∈ V be such that ♦m = ♦n. Then 1♦m = 1♦n so that m∨ 1♦ = n∨ 1♦. Hence m = n. Also, for any ♢ ∈ PC(V), if m = 1♢, then ρ♦m = m ∨ ρ♦ = 1♢ ∨ ρ♦ = ρ♢ by Lemma 14(5). Then ♢ is same as ♦m and m is a minimal element. Thus m ↣ ρ♦m is a bijection of M onto PC(V ). Theorem 12. If V is a PDL with two parapseudo-complementations ♦ and ♢, then the map f : V ♦ → V ♢ defined by f(ρ♦) = ρ♢ is an isomorphism of Boolean algebras. Proof. Let V be a PDL with two parapseudo-complementations ♦ and ♢. Clearly the map f : V ♦ → V ♢ defined by f(ρ♦) = ρ♢ is well -defined and one -one by Lemma 14. By definition, f is onto. Let ρ♦, ϱ♦ ∈ V . Then, f(ρ♦∧ϱ♦) = f((ρ♦ ♦ ∨ ϱ♦ ♦ )♦) = REFERENCES 1144 (ρ♦ ♦ ∨ ϱ♦ ♦ )♢ = (ρ ∨ ϱ)♦♦♢ = (ρ ∨ ϱ)♢♢♢ = (ρ♢ ♢ ∨ ϱ♢ ♢ )♢ = ρ♢∧ϱ♢ = f(ρ♦)∧f(ϱ♦). Also, f(ρ♦ ∨ ϱ♦) = f((ρ ∧ ϱ)♦) = (ρ ∧ ϱ)♢ = ρ♢ ∨ ϱ♢ = f(ρ♦) ∨ f(ϱ♦). Therefore f is an isomorphism. 6. Conclusions In this paper, we have introduced the concept of a parapseudo-complementation on a paradistributive lattiocoid and examined its elementary properties. By establishing necessary conditions, we have provided insights into when a PDL with a minimal element can be parapseudo-complemented. Furthermore, our investigation has focused on the equationally definable nature of parapseudo-complementation, identifying the properties required for this concept to be equationally definable within a PDL. This contributes to a deeper understanding of the formalization and algebraic implications of parapseudo- complementation. Additionally, we have established a one-to-one correspondence between the set of all minimal elements and the set of all parapseudo-complementations in a PDL. This correspondence highlights the interplay between minimal elements and parapseudo- complementation, providing a valuable connection between the structural elements of a PDL and the concept under study. In future, our work will focus on ♦-PDL, Stone PDL and study their topological properties. Conflicts of interest or competing interests The authors declare that they have no conflicts of interest. Informed Consent The authors are fully aware and satisfied with the contents of the article. Acknowledgements The authors wish to thank the anonymous reviewers for their valuable suggestions. References [1] R Bandaru and S Ajjarapu. Paradistributive latticoids. European Journal of Pure and Applied Mathematics, In Press. https://doi.org/10.29020/nybg.ejpam.v17i2.5042. [2] G Birkhoff. Lattice Theory. Colloquium Publications, American Mathematical Soci- ety, New York, 1940. [3] I Chajda and H Langer. Filters and congruences in sectionally pseudocomplemented lattices and posets. Soft Computing, 25:8827–8837, 2021. REFERENCES 1145 [4] I Chajda and H Langer. Implication in finite posets with pseudocomplemented sec- tions. Soft Computing, 26:5945–5953, 2022. [5] O Frink. Pseudo-complements in semi-lattices. Duke Mathematical Journal, 29:505– 514, 1962. [6] M Mandelker. Relative annhilators in lattices. Duke Mathematical Journal, 37:377– 386, 1970. [7] M Sambasiva Rao. δ-ideals in pseudo-complemented distributive lattices. Archivum Mathematicum, 48(2):97–105, 2012. [8] M Sambasiva Rao and A E Badawy. Normal ideals of pseudo-complemented distribu- tive lattices. Chamchuri Journal of Mathematics, 9:61–73, 2017. [9] E G Rezk. Closed ideals and annihilators of distributive dual weakly complemented lattice. European Journal Of Pure and Applied Mathematics, 15(2):486–495, 2022. [10] P Ribenboim. Characterization of the pseudo-complement in a distributive lattice with least element. Summa Brasiliensis Mathematicae, 2(4):43–49, 1949. [11] U M Swamy and G C Rao. Almost distributive lattices. Journal of the Australian Mathematical Society. Series A., 31:77–91, 1981. [12] U M Swamy, G C Rao, and G N Rao. Pseudo-complementation on almost distributive lattices. Southeast Asian Bulletin of Mathematics, 24:95–104, 2000. [13] C Thomaz. Dicomplemented Lattices. A contextual Generalization of Boolean Algebra. PhD thesis, TU Dresden, 2004. [14] P V Venkatanarasimhan. Pseudo-complements in posets. Proceedings of the American Mathematical Society, 28(1):9–17, 1971. [15] R Wille. Boolean concept logic. In B. Ganter and G. W. Mineau, editors, Conceptual Structures: Logical, Linguistic and Computational Issues., pages 14–18, Germany, 2000. ICCS 2000 Darmstadt.