EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 819-834 ISSN 1307-5543 – ejpam.com Published by New York Business Global Paradistributive Latticoids Ravikumar Bandaru1, Suryavardhani Ajjarapu2,∗ 1 Department of Mathematics, School of Advanced Sciences, VIT-AP University, Andhra Pradesh-522237, India 2 Department of Mathematics, GITAM Deemed to be University, Hyderabad Campus, Telangana-502329, India Abstract. In this paper, the concept of Paradistributive Latticoid (PDL) as a generalization of a distributive lattice is introduced and investigated its properties. A set of equivalent conditions for a PDL to become a distributive lattice are given. The notions of an ideal and a filter in a PDL are introduced and studied their properties. Subdirect representation of a PDL is obtained. 2020 Mathematics Subject Classifications: 06D99 Key Words and Phrases: Paradistributive Latticoid(PDL), Ideal, Filter, Congruence, Subdi- rectly irreducible. 1. Introduction Garret Birkhoff’s effort in the mid-1930’s arouse off with the overall development of lattice theory (see [1]). In a great sequence of works, he signified the circumstances of lattice theory and demonstrated how it provides a conjoining for independent advance- ments in many arithmetic disciplines. Birkhoff system is an algebra in which two binary operations meet and join, each of which is commutative, associative, and idempotent, and which, when combined, satisfy the relation x ∧ (x ∨ y) = x ∨ (x ∧ y) (see [3, 4]). This is a weakend version of the absorption law for lattices and was introduced in the year 1948. Lattices and quasilattices are examples of Birkhoff systems, with the latter being the regularization of a variety of lattices. The types of Birkhoff systems that adhere to one or both distributive rules were studied. They were three Birkhoff systems namely meet-distributive, join-distributive, and distributive. The duality between meet and join operations lead to corresponding results for join-distributive Birkhoff systems. A charac- terisation of these varieties, subvarieties, a duality thesis for distributive Birkhoff systems, a structure theory for meet-distributive Birkhoff systems improved descriptions of some of the subdirectly irreducibles. As one of the standard application of Birkhoff theorem ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5125 Email addresses: ravimaths83@gmail.com (R. Bandaru), syerrapr@gitam.in (S. Ajjarapu) https://www.ejpam.com 819 © 2024 EJPAM All rights reserved. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 820 is Stone’s theorem, here our aim of this paper is to use Birkhoff theorem for subdirectly irreducible PDLs. George Boole’s attempt to formulate propositional logic in the first part of the nine- teenth century induced the idea of Boolean algebras (see [2]). At the end of the course, he looked into the axiomatic of Boolean algebras. Distributive lattices had a main role in lat- tice theory. As lattice theory began with Boolean algebra, the postulation of distributive lattices is the most comprehensive with fulfilling chapter in the history of lattice theory. Many lattice conditions, as well as lattice components and ideals, are debilitate variants of distributivity. As a result, a detailed mastery of distributive lattices is required to perform in lattice theory. Distributive lattices are characterized by their lattice of ideals. Finally, in several applications, the distributivity constraint is enforced on lattices arising in various fields of mathematics, particularly algebra. Certain algebras are referred as distributive quasi lattices, and they are used to generalize distributive lattices. J. A. Kalman given subdirect decomposition of distributive quasilattices (see [5]) U. M. Swamy and G. C. Rao introduced the concept of an almost distributive lat- tice(ADL) (see [8]). This group of ADLs covers nearly all the existing ring theoretical hypothesis of a Boolean algebra. The class of triple systems has been introduced by Sub- rahmanyam as a lattice theoretic generalisation of P1-rings (see [6, 7]). For most of the results that are valid in triple systems the additive semigroup structure in the triple system does not play any role. This motivated them to introduce the class of almost distributive lattices. An ADL is an algebra (L,∨,∧) which fulfills all of the distributive lattice’s axioms with 0 except the possibility of commutativity with respect to ∨ and ∧. The idea of our paper is to introduce a new algebraic structure as a generalization of distributive lattice and distributive Birkhoff systems. We introduce Paradistributive Latticoids, which is an algebra of type (2,2,0) that assures all the propositions of a distributive lattice with the possible exception of the left distributivity of the operation meet and the commutativity of the operations join and meet. In §2, we discuss the basic definition of a PDL and some preliminary results related to PDL and illustrate few examples. In §3, we introduce the notions of an ideal and a filter in a PDL and investigate its properties. Further in §4, we also provide a subdirect representation theorem for associative PDLs using Birkhoff theorem, which simplifies many results in PDLs. 2. Paradistributive Latticoids In this section, we define a Paradistributive Latticoid and present some fundamental findings, the most of which require just simple verification. Definition 1. An algebra (V,∨,∧, 1) of type (2,2,0) is called a Paradistributive Latticoid, if it assures the subsequent axioms: (LD∨) p1 ∨ (p2 ∧ p3) = (p1 ∨ p2) ∧ (p1 ∨ p3). (RD∨) (p1 ∧ p2) ∨ p3 = (p1 ∨ p3) ∧ (p2 ∨ p3). (L1) (p1 ∨ p2) ∧ p2 = p2. (L2) (p1 ∨ p2) ∧ p1 = p1. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 821 (L3) p1 ∨ (p1 ∧ p2) = p1. (I1) p1 ∨ 1 = 1. for all p1, p2, p3 ∈ V . The independence of the axioms mentioned in the above definition can be verified us- ing non-trivial examples. The example below demonstrates how every non-empty set can be transformed into a PDL with any member that has been arbitrarily preassigned as its unity(greatest) element. For any p1, p2 ∈ V , we say that p1 is less than or equal to p2 and write p1 ≤ p2 if p1 ∧ p2 = p1 or equivalently p1 ∨ p2 = p2 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 y0 ∈ V . Then, for any x, y ∈ V define ∨ and ∧ on V by x ∨ y = { x y ̸= y0 y0 y = y0 and x ∧ y = { y y ̸= y0 x y = y0 Then (V,∨,∧, y0) is a disconnected PDL with y0 as its greatest element. Example 2. Let V = {0, 1, 2, 3, 4} be a set with binary operations ∨ and ∧ given in the following tables: ∨ 0 1 2 3 4 0 0 1 0 3 3 1 1 1 1 1 1 2 2 1 2 4 4 3 3 1 3 3 3 4 4 1 4 4 4 ∧ 0 1 2 3 4 0 0 0 2 0 2 1 0 1 2 3 4 2 0 2 2 0 2 3 0 3 2 3 4 4 0 4 2 3 4 Then (V ; ∨ ∧ 1) is a Paradistributive Latticoid. But V is not a distributive lattice, since 0 ∧ (2 ∨ 1) = 0 ∧ 1 = 0 ̸= 2 = 2 ∨ 0 = (0 ∧ 2) ∨ (0 ∧ 1), 0 ∧ 2 = 2 ̸= 0 = 2 ∧ 0 and 0 ∨ 2 = 0 ̸= 2 = 2 ∨ 0. Example 3. Let (V,+, ·, 0, 1) be a commutative regular ring with unity and let x0 be the unique idempotent element in V such that xV = x0V . Now, for any x, y ∈ V , define (1) x ∨ y = y0x (2) x ∧ y = x+ y − y0x. Then (V,∨,∧, 0) is a PDL. We now provide some essential results. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 822 Lemma 1. For any p1, p2 ∈ V , the following holds: (1) p1 ∧ p1 = p1. (2) p1 ∨ p1 = p1. (L4) (p1 ∧ p2) ∨ p2 = p2. (L5) p1 ∨ (p2 ∧ p1) = p1. (L6) p1 ∧ (p1 ∨ p2) = p1. Proof. Proofs of (1), (2) are obvious. Now (p1 ∧ p2)∨ p2 = (p1 ∨ p2)∧ (p2 ∨ p2) = (p1 ∨ p2)∧p2 = p2, which proves (L4). Also p1∨(p2∧p1) = (p1∨p2)∧(p1∧p1) = (p1∨p2)∧p1 = p1 which proves (L5). Lastly, (L6) follows as p1∧(p1∨p2) = (p1∨p1)∧(p1∨p2) = p1∨(p1∧p2) = p1. Lemma 2. For any p1, p2 ∈ V , p1 ∨ p2 = p2 ∨ p1 whenever p1 ≤ p2. Proof. Let p1, p2 ∈ V and p1 ≤ p2. Then p2 ∨ p1 = p1 = p1 ∨ (p2 ∧ p1) = p1 ∨ p2. Lemma 3. The relation ≤ is a partial ordering on V . Proof. The reflexivity of ≤ follows from Lemma 1. Let p1, p2 ∈ V be such that p1 ≤ p2 and p2 ≤ p1. That is p1 ∨ p2 = p2 and p2 ∨ p1 = p1 and hence by Lemma 2, we have p1 = p2. Thus ≤ is anti symmetric. Finally, suppose p1, p2, p3 ∈ V such that p1 ≤ p2 ≤ p3. Then p1 ∨ p3 = (p1 ∧ p2) ∨ p3 = (p1 ∨ p3) ∧ (p2 ∨ p3) = p3 ∧ p3 = p3 implies p1 ≤ p3, hence ≤ is transitive. Lemma 4. For any p1, p2 ∈ V , p1 ∧ p2 = p1 if and only if p1 ∨ p2 = p2. Proof. Let p1 ∧ p2 = p1. Then p1 ∨ p2 = (p1 ∧ p2)∨ p2 = p2. Similarly, for p1 ∨ p2 = p2, we have, p1 ∧ p2 = p1 ∧ (p1 ∨ p2) = p1. Lemma 5. For any p1, p2 ∈ V , p1 ∨ p2 = p1 if and only if p1 ∧ p2 = p2. Proof. Let p1∨p2 = p1. Then p1∧p2 = (p1∨p2)∧p2 = p2. Similarly, for p1∧p2 = p2, we have, p1 ∨ p2 = p1 ∨ (p1 ∧ p2) = p1. Lemma 6. For any p1, p2 ∈ V , the following holds: (1) (p1 ∨ p2) ∨ p2 = p1 ∨ p2. (2) (p1 ∨ p2) ∨ p1 = p1 ∨ p2. (3) p1 ∨ (p1 ∨ p2) = p1 ∨ p2. (4) p1 ∧ (p1 ∧ p2) = p1 ∧ p2. (5) (p1 ∧ p2) ∧ p2 = p1 ∧ p2. (6) p2 ∧ (p1 ∧ p2) = p1 ∧ p2. Proof. (1). By definition 1, we have (p1 ∨ p2) ∧ p2 = p2. Now, by Lemma 5, we have (p1 ∨ p2) ∨ p2 = p1 ∨ p2. (2). By definition 1, we have (p1∨p2)∧p1 = p1. Now, by Lemma 5, we have (p1∨p2)∨p1 = p1 ∨ p2. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 823 Lemma 7. For any p1 ∈ V , we have (I3) p1 ∧ 1 = p1. (I4) 1 ∧ p1 = p1. (I5) 1 ∨ p1 = 1. Proof. Let p1 ∈ V . Then p1 ∧ 1 = p1 ∧ (p1 ∨ 1) = p1 and 1 ∧ p1 = (p1 ∨ 1) ∧ p1 = p1. Similarly, 1 ∨ p1 = 1 ∨ (1 ∧ p1) = 1 which proves (I5). Theorem 1. For any p1, p2, p3 ∈ V , (RD∧) (p1 ∨ p2) ∧ p3 = (p1 ∧ p3) ∨ (p2 ∧ p3). Proof. Let p1, p2, p3 ∈ V . Write x = (p1 ∨ p2) ∧ p3 and y = (p1 ∧ p3) ∨ (p2 ∧ p3) = [(p1 ∧ p3) ∨ p2] ∧ p3. Then, by (RD∨), (LD∨), (L3), (L5) and (L6), we have x ∨ y = [(p1 ∨ p2) ∧ p3] ∨ [(p1 ∧ p3) ∨ (p2 ∧ p3)] = [(p1 ∨ p2) ∨ [(p1 ∧ p3) ∨ (p2 ∧ p3)]] ∧ p3 = [(p1 ∨ p2) ∨ [((p1 ∧ p3) ∨ p2) ∧ p3]] ∧ p3 = [(p1 ∨ p2) ∨ [((p1 ∨ p2) ∧ (p3 ∨ p2)) ∧ p3]] ∧ p3 = [(p1 ∨ p2) ∧ [(p1 ∨ p2) ∨ p3]] ∧ p3 = [(p1 ∨ p2) ∧ p3] = x Also, x ∨ y = [(p1 ∨ p2) ∧ p3] ∨ [((p1 ∧ p3) ∨ p2) ∧ p3] = [((p1 ∨ p2) ∨ ((p1 ∧ p3) ∨ p2)) ∧ (p3 ∨ ((p1 ∧ p3) ∨ p2))] ∧ p3 = [((p1 ∨ p2) ∨ ((p1 ∨ p2) ∧ (p3 ∨ p2))) ∧ (p3 ∨ ((p1 ∨ p2) ∧ (p3 ∨ p2)))] ∧ p3 = [(p1 ∨ p2) ∧ ((p3 ∨ (p1 ∨ p2)) ∧ (p3 ∨ p2))] ∧ p3 = [(p1 ∨ p2) ∧ (p3 ∨ ((p1 ∨ p2) ∧ p2))] ∧ p3 = [(p1 ∨ p2) ∧ (p3 ∨ p2)] ∧ p3 = [(p1 ∧ p3) ∨ p2] ∧ p3. = y Therefore, (p1 ∨ p2) ∧ p3 = (p1 ∧ p3) ∨ (p2 ∧ p3). Theorem 2. For any p1, p2 ∈ V , the following are equivalent: (1) (p1 ∧ p2) ∨ p1 = p1. (2) p1 ∧ (p2 ∨ p1) = p1. (3) (p2 ∧ p1) ∨ p2 = p2. (4) p2 ∧ (p1 ∨ p2) = p2. (5) p1 ∧ p2 = p2 ∧ p1. (6) p1 ∨ p2 = p2 ∨ p1. (7) p1 ∧ p2 ≤ p1. (8) There exists a ∈ V such that a ≤ p1 and a ≤ p2. (9) The g.l.b of p1 and p2 exists in V and equals p1 ∧ p2. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 824 Proof. Let p1, p2 ∈ V . (1) ⇒ (2) : Assume (1). Then p1 ∧ (p2 ∨ p1) = (p1 ∨ p1) ∧ (p2 ∨ p1) = (p1 ∧ p2) ∨ p1 = p1. (2) ⇒ (1) : Assume (2). Then (p1 ∧ p2) ∨ p1 = (p1 ∨ p1) ∧ (p2 ∨ p1) = p1 ∧ (p2 ∨ p1) = p1. (3) ⇒ (4) : Assume (3). Then p2 ∧ (p1 ∨ p2) = (p2 ∨ p2) ∧ (p1 ∨ p2) = (p2 ∧ p1) ∨ p2 = p2 (4) ⇒ (3) : Assume (4). Then (p2 ∧ p1) ∨ p2 = (p2 ∨ p2) ∧ (p1 ∨ p2) = p2 ∧ (p1 ∨ p2) = p2. (1) ⇒ (5) : Assume (1). Then p2 ∧ p1 = [(p1 ∧ p2) ∨ p2] ∧ [(p1 ∧ p2) ∨ p1] = (p1 ∧ p2) ∨ (p2 ∧ p1) = [(p1 ∨ (p2 ∧ p1)] ∧ [(p2 ∨ (p2 ∧ p1)] = p1 ∧ p2. (5) ⇒ (1) : Assume (5). Then (p1 ∧ p2) ∨ p2 = (p2 ∧ p1) ∨ p2 = p1. (5) ⇒ (3) : Assume (5). Then (p2 ∧ p1) ∨ p2 = (p1 ∧ p2) ∨ p2 = p2. (3) ⇒ (5) : Assume (3). Then p1 ∧ p2 = [(p2 ∧ p1) ∨ p1] ∧ [(p2 ∧ p1) ∨ p2] = [p2 ∧ p1) ∧ [(p2 ∧ p1) ∨ p2]] ∨ [p1 ∧ [(p2 ∧ p1) ∨ p2)] = (p2 ∧ p1) ∨ (p1 ∧ p2) = [p2 ∨ (p1 ∧ p2)] ∧ [p1 ∨ (p1 ∧ p2)] = p2 ∧ p1. (6) ⇒ (3) : Assume (6). Then p1 ∧ (p2 ∨ p1) = p1 ∧ (p1 ∨ p2) = p1. (3) ⇒ (6) : Assume (3). Then p2 ∨ p1 = p2 ∨ ((p1 ∨ p2) ∧ p1) = (p2 ∨ (p1 ∨ p2)) ∧ (p2 ∨ p1) = (p1 ∨ p2) ∧ (p2 ∨ p1) = (p1 ∨ p2) ∧ (p1 ∨ (p2 ∨ p1)) = p1 ∨ (p2 ∧ (p2 ∨ p1)) = p1 ∨ p2. Now we prove the equivalence of (5), (7), (8) and (9). (5) ⇒ (7) : Assume (5). Then p1 ∧ p2 = p2 ∧ p1 ≤ p1. Hence (7) follows. (7) ⇒ (8) : Assume (7). Then p1 ∧ p2 ≤ p1 and let a = p1 ∧ p2. Therefore a ≤ p1 and a ≤ p2. (8) ⇒ (9) : Assume (8). Then there exists a ∈ V such that a ≤ p1 and a ≤ p2. Now consider (p1 ∧ p2) ∨ p1 = (p1 ∨ p1) ∧ (p2 ∨ p1) = p1 ∧ (p2 ∨ p1) = (a ∨ p1) ∧ (p2 ∨ p1) = (a ∧ p2) ∨ p1 = a ∨ p1 = p1 Therefore p1 ∧ p2 is the lower bound of p1 and p2 in V . Now, for c ∈ V such that c ≤ p1 and c ≤ p2, we have c ∨ (p1 ∧ p2) = (c ∨ p1) ∧ (c ∨ p2) = p1 ∧ p2, implies c ≤ p1 ∧ p2. Thus R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 825 p1 ∧ p2 is the g.l.b of p1 and p2 in V . (9) ⇒ (5): Assume (9). Then (p1 ∧ p2) ∨ p1 = p1, since p1 ∧ p2 ≤ p1 and hence (5) follows by the equivalence of (1) and (5). Thus the conditions (5), (7), (8) and (9) are equivalent. Theorem 3. For any p1, p2, p3 ∈ V , p1 ∨ (p2 ∧ p3) = (p1 ∨ p2) ∧ (p3 ∨ p1) if and only if p1 ∧ p2 = p2 ∧ p1. Proof. Let p1, p2, p3 ∈ V be such that p1 ∨ (p2 ∧ p3) = (p1 ∨ p2) ∧ (p3 ∨ p1). Then p2 ∧ p1 = ((p1 ∧ p2) ∨ p2) ∧ ((p1 ∨ (p1 ∧ p2)) = (p1 ∧ p2) ∨ (p2 ∧ p1) = (p1 ∨ (p2 ∧ p1)) ∧ (p2 ∨ (p2 ∧ p1)) = p1 ∧ p2. Converse follows from Theorem 2 and (LD∨). Lemma 8. For any p1, p2, p3 ∈ V , p1 ∨ (p2 ∧ p3) = p1 ∨ (p3 ∧ p2). Proof. Since p1 ≤ p1 ∨ p2 and p1 ≤ p1 ∨ p3, we have (p1 ∨ p2) ∧ (p1 ∨ p3) = (p1 ∨ p3) ∧ (p1 ∨ p2). Therefore p1 ∨ (p2 ∧ p3) = (p1 ∨ p2) ∧ (p1 ∨ p3) = (p1 ∨ p3) ∧ (p1 ∨ p2) = p1 ∨ (p3 ∧ p2). Theorem 4. The operation ∨ is associative in a PDL V . Proof. Let p1, p2, p3 ∈ V . Then p1 ∨ (p2 ∨ p3) = [p1 ∨ (p3 ∧ p1)] ∨ (p2 ∨ p3) = [((p1 ∨ p2) ∧ p1) ∨ (p3 ∧ p1)] ∨ (p2 ∨ p3) = [[(p1 ∨ p2) ∨ p3] ∧ p1] ∨ (p2 ∨ p3) = [[(p1 ∨ p2) ∨ p3] ∨ (p2 ∨ p3)] ∧ [p1 ∨ (p2 ∨ p3)] = ((p1 ∨ p2) ∨ p3) ∧ (p1 ∨ (p2 ∨ p3)) = ((p1 ∨ p2) ∧ (p1 ∨ (p2 ∨ p3))) ∨ (p3 ∧ (p1 ∨ (p2 ∨ p3))) = (p1 ∨ p2) ∨ ((p1 ∨ (p2 ∨ p3)) ∧ p3) = (p1 ∨ p2) ∨ [(p1 ∧ p3) ∨ ((p2 ∨ p3) ∧ p3)] = (p1 ∨ p2) ∨ [(p1 ∧ p3) ∨ p3] = (p1 ∨ p2) ∨ p3 Therefore ∨ is associative in V . “Note that, since ∨ is associative in a PDL, we can write (p1 ∨ p2) ∨ p3, simply, as p1 ∨ p2 ∨ p3. Hence the notation ( n ∨ i=1 αi) is meaningful, as well”. The operation ∧ is not associative, as shown by the following example: R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 826 Example 4. Let V = {0, 1, 2, 3, 4, 5} be a set with binary operations ∨ and ∧ given in the following tables: ∨ 0 1 2 3 4 5 0 0 1 4 0 4 0 1 1 1 1 1 1 1 2 2 1 2 2 2 2 3 3 1 2 3 2 3 4 4 1 4 4 4 4 5 5 1 4 5 4 5 ∧ 0 1 2 3 4 5 0 0 0 3 3 0 5 1 0 1 2 3 4 5 2 0 2 2 3 4 5 3 0 3 3 3 5 5 4 0 4 2 3 4 5 5 0 5 3 3 5 5 Then (V ; ∨ ∧ 1) is a Paradistributive Latticoid. But the operation ∧ is not associative, since 0 ∧ (2 ∧ 4) = 0 ∧ 4 = 0 ̸= 5 = 3 ∧ 4 = (0 ∧ 2) ∧ 4. Definition 2. A Paradistributive Latticoid (V,∨,∧, 1) is said to be associative if it satisfies the following condition p1 ∧ (p2 ∧ p3) = (p1 ∧ p2) ∧ p3 for all p1, p2, p3 ∈ V. Example 5. Let V = {0, 1, 2, 3, 4} be a set with binary operations ∨ and ∧ given in the following tables: ∨ 0 1 2 3 4 0 0 1 0 3 3 1 1 1 1 1 1 2 2 1 2 4 4 3 3 1 3 3 3 4 4 1 4 4 4 ∧ 0 1 2 3 4 0 0 0 2 0 2 1 0 1 2 3 4 2 0 2 2 0 2 3 0 3 2 3 4 4 0 4 2 3 4 Then (V ; ∨ ∧ 1) is an associative Paradistributive Latticoid. Lemma 9. For any p1, p2, p3 ∈ V , p1 ∨ p2 ∨ p3 = p1 ∨ p3 ∨ p2. Proof. Since p1 ≤ p1 ∨ p2 and p1 ≤ p1 ∨ p3, we have (p1 ∨ p2) ∨ (p1 ∨ p3) = (p1 ∨ p3) ∨ (p1 ∨ p2). Therefore (p1 ∨ p2 ∨ p1) ∨ p3 = (p1 ∨ p3 ∨ p1) ∨ p2. Hence p1 ∨ p2 ∨ p3 = p1 ∨ p3 ∨ p2. Lemma 10. For any p1, p2 ∈ V , p1 ∨ p2 = 1 if and only if p2 ∨ p1 = 1. Theorem 5. Let (V,∨,∧, 1) be a PDL. Then the following are equivalent: (1) (V,∨,∧, 1) is a distributive lattice. (2) The poset (V,≤) is directed below. (3) (p1 ∧ p2) ∨ p1 = p1. (4) The operation ∧ is commutative. (5) The operation ∨ is commutative. (6) The relation χ = {(p1, p2) ∈ V × V | p2 ∨ p1 = p2} is antisymmetric. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 827 Proof. (1) ⇒ (2): Let (V,∨,∧, 1) be a distributive lattice. Then, by Theorem 2, V is directed below. The equivalence of (2), (3), (4), (5) also follows from Theorem 2. (5) ⇒ (6): Given χ = {(p1, p2) ∈ V × V | p2 ∨ p1 = p2}. If (p1, p2) ∈ χ, then p2 ∨ p1 = p1 and (p2, p1) ∈ χ and hence p1 = p2. (6) ⇒ (1): Let p1, p2 ∈ V . Then (p1∨p2)∨(p2∨p1) = p1∨p2∨p1 = p1∨p1∨p2 = p1∨p2 and hence (p1∨p2, p2∨p1) ∈ χ. Also, (p2∨p1)∨ (p1∨p2) = p2∨p1∨p2 = p2∨p2∨p1 = p2∨p1 which shows (p2 ∨ p1, p1 ∨ p2) ∈ χ. Since χ is antisymmetric, we have p1 ∨ p2 = p2 ∨ p1. So that V is a lattice and hence distributive. Theorem 6. An algebra (V,∨,∧, 1) of type (2, 2, 0) is a PDL if and only if it satisfies the following: (LD∨) p1 ∨ (p2 ∧ p3) = (p1 ∨ p2) ∧ (p1 ∨ p3) (RD∨) (p1 ∧ p2) ∨ p3 = (p1 ∨ p3) ∧ (p2 ∨ p3) (RD∧) (p1 ∨ p2) ∧ p3 = (p1 ∧ p3) ∨ (p2 ∧ p3) (L1) (p1 ∨ p2) ∧ p2 = p2 (L3) p1 ∨ (p1 ∧ p2) = p1 (I1) p1 ∨ 1 = 1 (I2) 1 ∧ p1 = p1. for all p1, p2, p3 ∈ V . Corollary 3. For all a ∈ V , V contains an element 0 such that 0 ∨ a = a if and only if V is a bounded distributive lattice, and hence for any a ∈ V , the set Va = {a ≤ p1 | p1 ∈ V } is a bounded distributive lattice under the induced operations ∨ and ∧ with a as its least element. 3. Ideals and Filters In this section, we introduce the notion of an ideal and a filter in a paradistributive latticoid and investigate its important properties. Definition 4. A non-empty subset U of V is said to be an ideal if it satisfies the following: p1, p2 ∈ U ⇒ p1 ∨ p2 ∈ U. p1 ∈ U, a ∈ V ⇒ p1 ∧ a ∈ U. It should be noted that every ideal of V is a PDL. Next theorem describes the ideal generated by a non-empty subset S of V . Theorem 7. Let S be a non-empty subset of V . Then (S] = {( n ∨ i=1 αi) ∧ s | αi ∈ S, s ∈ V and n is a positive integer } is the smallest ideal of V containing S. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 828 Proof. Let S be a non-empty subset of V . Choose T = { n ∨ i=1 αi | αi ∈ S for 1 ≤ i ≤ n and n is a positive integer } Clearly S ⊆ T ⊆ (S]. First we prove that (S] = {p1 ∈ V | t ∨ p1 = t for some t ∈ T} =M. Let x ∈ (S]. Then x = ( n ∨ i=1 αi) ∧ s where αi ∈ S, s ∈ V . Now ( n ∨ i=1 αi) ∨ x = ( n ∨ i=1 αi) ∨ (( n ∨ i=1 αi) ∧ s). = n ∨ i=1 αi which implies x ∈ M . Therefore (S] ⊆ M . Conversely, let s ∈ M . Then t ∨ s = t for some t = n ∨ i=1 αi, αi ∈ S. Now s = t ∧ s = ( n ∨ i=1 αi) ∧ s ∈ (S]. Therefore M ⊆ (S]. Hence (S] = {s ∈ V | t ∨ s = t for some t ∈ T} =M. Let s, l ∈ (S], then there exists t1, t2 ∈ T such that t1 ∨ s = t1 and t2 ∨ l = t2. Then (S] is an ideal as (t1 ∨ t2) ∨ (s ∨ l) = t1 ∨ (t2 ∨ (s ∨ l)) = t1 ∨ (t2 ∨ l ∨ s) = t1 ∨ t2 ∨ s = t1 ∨ s ∨ t2 = t1 ∨ t2 Therefore s∨ l ∈ (S]. Also, for s ∈ (S] and u ∈ V , we have t ∈ T such that t∨ s = t. Now t ∨ (s ∧ u) = (t ∨ s) ∧ (t ∨ u) = t ∧ (t ∨ u) = t. Hence s ∧ u ∈ (S]. Thus (S] is an ideal of V containing S. Now, let U be any ideal of V such that S ⊆ U . Let s ∈ (S]. Then s = ( n ∨ i=1 αi) ∧ p2 where αi ∈ S ⊆ U for 1 ≤ i ≤ n and p2 ∈ V . Since U is an ideal of V , we have s = ( n ∨ i=1 αi) ∧ p2 ∈ U. Hence (S] ⊆ U . Therefore (S] is the smallest ideal of V containing S. Note that if S = {a}, then we write (S] = (a], the principal ideal of V generated by ‘a’. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 829 Corollary 5. p1 ∈ (p2] if and only if p2 ∧ p1 = p1, where p1, p2 ∈ V . Lemma 11. Let U be an ideal of V . Then, for any p1, p2 ∈ V , p1 ∧ p2 ∈ U if and only if p2 ∧ p1 ∈ U . Corollary 6. For any p1, p2 ∈ V (p1] ∨ (p2] = (p1 ∨ p2] = (p2 ∨ p1] (p1] ∧ (p2] = (p1 ∧ p2] = (p2 ∧ p1] Theorem 8. The set P (V ) of all ideals of V forms a distributive lattice with greatest element under the set inclusion in which the g.l.b and l.u.b for any P and Q are respectively, P ∧Q = P ∩Q and P ∨Q = {p1 ∨ p2 | p1 ∈ P, p2 ∈ Q}. Definition 7. A non-empty subset F of V is said to be a filter if it satisfies the following: p1, p2 ∈ F ⇒ p1 ∧ p2 ∈ F. p1 ∈ F, a ∈ V ⇒ a ∨ p1 ∈ F. Theorem 9. Let S be a non-empty subset of V . Then [S) = {p1 ∨ ( n ∧ i=1 si) | si ∈ S, p1 ∈ V, 1 ≤ i ≤ n and n is a positive integer } is the smallest filter of V containing S. Proof. Let a, b ∈ [S). Then a = p1 ∨ ( n ∧ i=1 si), b = p2 ∨ ( m ∧ i=1 tj) a ∧ b = [p1 ∨ ( n ∧ i=1 si)] ∧ [p2 ∨ ( m ∧ j=1 tj)] = [p1 ∧ (p2 ∨ ( m ∧ j=1 tj))] ∨ [( n ∧ i=1 si) ∧ (p2 ∨ ( m ∧ j=1 tj))] = [p1 ∧ (p2 ∨ ( m ∧ j=1 tj))] ∨ [(p2 ∨ ( m ∧ j=1 tj)) ∧ ( n ∧ i=1 si)] = p3 ∨ ( m ∧ j=1 tj ∧ n ∧ i=1 si) (where p3 = (p1 ∧ (p2 ∨ ( m ∧ j=1 tj))) ∨ (p2 ∧ ( n ∧ i=1 si)) and hence a ∧ b ∈ [S). Now, we prove u ∨ a ∈ [S) for u ∈ V . Consider u ∨ a = u ∨ (p1 ∨ ( n ∧ i=1 si)) = (u ∨ p1) ∨ ( n ∧ i=1 si) ∈ [S) Therefore [S) is a filter of V , and clearly it is the smallest filter of V containing S. Note that if S = {a}, then we write [S) = [a), the principal filter of V generated by ‘a’. Corollary 8. p1 ∈ [p2) if and only if p1 = p1 ∨ p2 for all p1, p2 ∈ V . R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 830 Lemma 12. Let F be a filter of V and p1, p2 ∈ V . Then (1) p1 ∨ p2 ∈ F if and only if p2 ∨ p1 ∈ F . (2) For any p1, p2 ∈ V, [p1 ∨ p2) = [p2 ∨ p1). (3) For any p1, p2 ∈ V, [p1 ∧ p2) = [p2 ∧ p1) = [p1) ∨ [p2). Theorem 10. Let p1, p2 ∈ V . Then the following are equivalent. (1) (p1] ⊆ (p2]. (2) p2 ∧ p1 = p1. (3) p2 ∨ p1 = p2. (4) [p2) ⊆ [p1). Theorem 11. The collection F (V ) 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 = {p1 ∧ p2 | p1 ∈ F and p2 ∈ G}. Theorem 12. The class PU(V )(PF (V )) of all principal ideals(filters) of V is a sublattice of the distributive lattice U(V )(F (V )) of all the ideals(filters) of V . Moreover, the lattice PU(V ) is “dually isomorphic” on to the lattice PF (V ). Proof. Let PU(V ) be the set of all principal ideals of the PDL V and PU(V ) = {(a] | a ∈ V }. First, we prove that PU(V ) is a sublattice of U(V ). Let (a], (b] ∈ PU(V ). Then (a]∨(b] = (a ∨ b] for a, b ∈ V . Hence (a] ∨ (b] ∈ PU(V ). Similarly, (a] ∧ (b] = (a ∧ b] ∈ PU(V ). Therefore PU(V ) is a sublattice of U(V ). Finally, we prove that there exists a dual isomorphism from PU(V ) to PF (V ). Define ζ : PU(V ) → PF (V ) by ζ{(a]} = [a), a ∈ V . (i) ζ is a homomorphism: ζ{(a] ∨ (b]} = ζ{(a ∨ b]} = [a ∨ b) = [a) ∧ [b) = ζ{(a]} ∧ ζ{(b]} also, ζ{(a] ∧ (b]} = ζ{(a ∧ b]} = [a ∧ b) = [a) ∨ [b) = ζ{(a]} ∨ ζ{(b]} (ii) ζ is one-one: Let a, b ∈ V . Then ζ{(a]} = ζ{(b]} ⇒ [a) = [b) ⇒ (a] = (b] R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 831 (iii) ζ is onto: Let b ∈ PF (V ). Then b = [a) for some a ∈ V . Hence (a] ∈ PU(V ). Therefore ζ((a]) = [a) = b. Therefore there exists a dual isomorphism from PU(V ) onto PF (V ). 4. Subdirectly Irreducible PDLs For any algebra A, we denote the structure lattice of A, that is the lattice of all con- gruence relations on A, by B in which the least element is ∆A where ∆A = {(x, y) ∈ A × A | x = y} and greatest element is A × A. Recall that a non-trivial algebra A is said to be subdirectly irreducible if intersection of any family of nonzero congruences is again non-zero; or equivalently, B has smallest non-zero congruence. We characterize subdirectly irreducible associative PDLs in this section, and then use Birkhoff’s subdirect representation theorem to obtain a subdirect representation for an associative PDL. This subdirect representation of a PDL V is crucial in the theory of PDLs since it simplifies numerous lattice theoretic computations. Definition 9. Let V be a PDL, an element a ∈ V is said to be minimal if for any u ∈ V , u ≤ a⇒ u = a. Lemma 13. Let V be a PDL. Then for any a ∈ V, the following are equivalent: (1). a is minimal (2). p1 ∧ a = a for all p1 ∈ V (3). p1 ∨ a = p1 for all p1 ∈ V . Definition 10. An equivalence relation θ on a PDL V , is called a congruence relation on V if (a ∧ c, b ∧ d), (a ∨ c, b ∨ d) ∈ θ, for all (a, b), (c, d) ∈ θ. Lemma 14. For any a ∈ V , φa = {(p1, p2) ∈ V × V | p1 ∨ a = p2 ∨ a} is a congruence relation on V . Further, φa = ∆V if and only if a is minimal element of V . Proof. Given φa = {(p1, p2) ∈ V × V | p1 ∨ a = p2 ∨ a} . Clearly φa is an equivalence relation on V . Let (p1, p2), (p3, s) ∈ φa. Then p1 ∨ a = p2 ∨ a and p3 ∨ a = s ∨ a. Hence (p1 ∨ p3) ∨ a = p1 ∨ p3 ∨ a = p1 ∨ s ∨ a = p1 ∨ a ∨ s = p2 ∨ a ∨ s = p2 ∨ s ∨ a and (p1 ∧ p3) ∨ a = (p1 ∨ a) ∧ (p3 ∨ a) = (p2 ∨ a) ∧ (s ∨ a) = (p2 ∧ s) ∨ a. R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 832 Therefore (p1 ∨ p3, p2 ∨ s),∈ φa and (p1 ∧ p3, p2 ∧ s) ∈ φa. Therefore φa is a congruence relation on V . Assume φa = ∆V . Let p1 ∈ V be such that p1 ≤ a. Then a ∨ a = a = p1 ∨ a which implies (a, p1) ∈ φa = ∆V . Therefore a = p1. Hence a is minimal element. Let (p1, p2) ∈ φa. Then p1 ∨ a = p2 ∨ a. Since a is minimal element, we have p1 = p2. Hence φ a = ∆V . Lemma 15. If V is an associative PDL, then for any a ∈ V , θa = {(p1, p2) ∈ V ×V | p1∧a = p2∧a} is a congruence relation on V . Further, θa = ∆V if and only if a is unity (greatest) element of V . Proof. Let θa = {(p1, p2) ∈ V × V | p1 ∧ a = p2 ∧ a}. Clearly θa is an equivalence relation on V . Let (p1, p2) ∈ θa and (p3, t) ∈ θa. Then p1 ∧ a = p2 ∧ a and p3 ∧ a = t ∧ a. Hence (p1 ∧ p3) ∧ a = p1 ∧ p3 ∧ a = p1 ∧ t ∧ a = p1 ∧ a ∧ t ∧ a = p2 ∧ a ∧ t ∧ a = p2 ∧ t ∧ a = (p2 ∧ t) ∧ a and (p1 ∨ p3) ∧ a = (p1 ∧ a) ∨ (p3 ∧ a) = (p2 ∧ a) ∨ (t ∧ a) = (p2 ∨ t) ∧ a. Therefore (p1 ∧ p3, p2 ∧ t), (p1 ∨ p3, p2 ∨ t) ∈ θa. Hence θa is a congruence relation on V . Now assume θa = ∆V . Then, for any p1 ∈ V , a ∧ a = a = (p1 ∨ a) ∧ a implies (a, p1 ∨ a) ∈ θa = ∆V which shows that a is unity element of V . Conversely, suppose that a is unity element of V . Then, for (p1, p2) ∈ θa, we have p1∧a = p2 ∧ a which implies p1 = p2. Therefore θa = ∆V , which concludes the lemma. Theorem 13. Let V be a subdirectly irreducible associative PDL. Then every non-unity of V is minimal and V contains atmost two non-unity elements. Proof. Let V be an associative PDL. Suppose V is subdirectly irreducible. Let θ1 be the smallest non-zero congruence on V . Select p1, p2 ∈ V with p1 ̸= p2 such that (p1, p2) ∈ θ1. Then atleast one of p1 and p2 is minimal. For if, assume that p1 and p2 both are not minimal elements of V , then φp1 ̸= ∆V ̸= φp2 , implies (p1, p2) ∈ φp1 ∩ φp2 . Since p1 = p1 ∨ p1 = p2 ∨ p1 and p2 = p1 ∨ p2. Thus p1 = p2, which is a contradiction. Thus, atleast one of p1, p2 is minimal. Without loss of generality, let us assume that p1 is minimal. Let a be a non-unity element of V . Suppose, a is not minimal, then p1 being the minimal element we have a∧ p1 = p1. Hence a∨ p1 = a∨ (a∧ p1) = a, so that p1 ∨ a is a non-unity element of V . Therefore θp1∨a ̸= ∆V . Hence (p1, p2) ∈ θp1∨a. Also, since a is not minimal, we obtain (p1, p2) ∈ φa. Now p1 = p1 ∧ (p1 ∨ a) = p2 ∧ (p1 ∨ a) = p2 ∧ (p2 ∨ a) = p2 which R. Bandaru, S. Ajjarapu / Eur. J. Pure Appl. Math, 17 (2) (2024), 819-834 833 is a contradiction. Thus a is minimal. Suppose a, b, c ∈ V be three distinct non-unity elements of V . Then since a, b, c are minimal elements, we have φ = ∆V ∪ {(a, b), (b, a)} and ψ = ∆V ∪ {(b, c), (c, b)} are two non-zero congruences on V such that φ ∩ ψ = ∆V which contradicts the fact that V is subdirectly irreducible. Hence V has atmost two non-unity elements. Corollary 11. A subdirectly irreducible distributive lattice V is a two element chain. Remark: It is evident that the converse of Theorem 13 is also true. i.e., if V is a PDL containing atmost two non-unity elements and every non-unity element of V is minimal. Theorem 14. Let V be a PDL. Then the following are equivalent: (1) V is associative. (2) θa is a congruence relation on V for all a ∈ V . (3) V is a subdirect product of PDLs in each of which there are atmost two non-unity elements and every non-unity element is minimal. Proof. Let V be a PDL. We have (1) ⇒ (2) by Lemma 15. Assume (2). Then, for any congruence η on V , p1, p2 and p3 ∈ V we have (p1 ∧ p3, p2 ∧ p3) ∈ η if and only if (p1, p2) ∈ θp3 ∧ η. Hence for any congruence relation η on V , the quotient PDL V |η also satisfies the assumption (2). By Birkhoff’s theorem and Theorem 13, we have (3). (3) ⇒ (1) is clear. 5. Conclusions In this paper, we have introduced the concept of Paradistributive Latticoid and studied certain properties related to the structure. Further, provided a set of equivalence condi- tions for the PDL to become a distributive lattice. We have introduced the notions of an ideal and a filter in a PDL and studied their properties. We have obtained subdirect representation of a PDL. In future, our work will focus on parapseudo-complementation on a PDL, Stone PDL, Normal PDL and study their topological properties. Conflicts of interest or competing interests The authors declare that they have no conflicts of interest. Data and code Availability No data were used to support this study Supplementary information Not Applicable REFERENCES 834 Ethical Approval This article does not contain any studies with human participants or animals performed by any of the authors 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] G Birkhoff. Lattice Theory. 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