Ratio Mathematica Volume 44, 2022 On the effect of doubling of intervals on the 0 - distributive property of the lattice of weak congruences of chains Gladys Mano Amirtha V1 D. Premalatha2 Abstract Alan Day’s doubling construction of intervals has been found to affect some properties of the lattice of weak congruences of chains. Here, in this paper, we study how it affects the property of 0-distributivity of the lattice of weak congruences of chains. Keywords: doubling construction in lattices, weak congruence lattices, 0-distributive lattices. 2010 AMS subject classification: 06B10, 06D993. 1Reg. No.: 20111172092013, Ph.D. Research Scholar (Full time), PG and Research Department of Mathematics, (Rani Anna Government College for Women, Tirunelveli-627008, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.) gladyspeter3@gmail.com. 2Associate Professor, PG and Research Department of Mathematics, (Rani Anna Government College for Women, Tirunelveli-627008, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.) lathaaedward@gmail.com. 3 Received on June 7th, 2022. Accepted on Aug 10th, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.888. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 42 mailto:gladyspeter3@gmail.com mailto:lathaaedward@gmail.com Gladys Mano Amirtha V & D. Premalatha 1. Introduction Vojvodić G. and Šešelja B. initiated the study of the concept of weak congruences of a lattice in the year 1988 [12]. J. C. Varlet [9] was the first to introduce the concept of 0-distributive lattices. Several other researchers made various contributions in different aspects of 0-distributivity. For example, one can refer to [1], [8], [11]. A. Veeramani [10] in his thesis, has studied about the lattice of weak congruences of a finite chain, Boolean lattices and the lattices 𝐶𝑛, 𝑀3, 𝑁5first by considering 0 and 1 as non-constants and then considering them as constants, again by considering the Boolean lattice as an algebra and he studied about some weaker properties like 0-distributivity, 0-modularity, consistency, etc. G. Gratzer constructed a new lattice LUfrom a given lattice L by adding an element aU called the double of a ≠ 0 or 1 in L where LU = L ∪ {aU}with a new order denoted by ≤U [6]. Following that construction, A Day [3] introduced a similar construction L[I]by doubling an interval I of a given lattice L. After that it witnessed many developments, e.g., see [4], [5], [7]. Alan Day in [2] proved that a distributive lattice remains distributive when it is doubled by either a lower interval or an upper interval. In our present study, we analyse the effect of doubling of intervals on the property of 0- distributivity in the lattice of weak congruences of chains. 2. Preliminaries Definition 2.1 [6] A lattice 𝐿 satisfying the following identities • 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) • 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝐿 is called a distributive lattice. If not, it is a non-distributive lattice. Definition 2.2 [4] A lattice 𝐿 is said to be 0 - distributive, if for all 𝑥, 𝑦, 𝑧 ∈ 𝐿, whenever 𝑥 ∧ 𝑦 = 0 and 𝑥 ∧ 𝑧 = 0, then𝑥 ∧ (𝑦 ∨ 𝑧) = 0. Lemma 2.3 [10] If 𝐿𝑛 is a chain of 𝑛 elements, then 𝐶𝑊(𝐿𝑛) is 0 - distributive. Definition 2.4 [6] An equivalence relation 𝜃 on a lattice 𝐿 is said to be a congruence relation on 𝐿, if it is compatible with both meet and join, that is, for all 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝐿, 𝑎 ≡ 𝑏 (𝜃) and 𝑐 ≡ 𝑑 (𝜃) imply that 𝑎 ∨ 𝑐 ≡ 𝑏 ∨ 𝑑 (𝜃) and 𝑎 ∧ 𝑐 ≡ 𝑏 ∧ 𝑑 (𝜃). Definition 2.5 [12] A weak congruence relation on an algebra 𝐴 is a symmetric and transitive sub-universe of 𝐴2. Note 2.6 The lattice of all weak congruence relations of 𝐿including 𝜙 with respect to the relation ⊆ is denoted by 𝐶𝑊(𝐿). We consider 0 and 1 of 𝐿 as non-constants in this paper. Remark 2.7 [12] In 𝐶𝑊(𝐿), we have • [𝜙, 𝛥] ≅ 𝑆𝑢𝑏(𝐿), the lattice of all sublattices of 𝐿. 43 On the effect of doubling of intervals on the 0-distributive property of the lattice of weak congruences of chains • [𝛥, 𝜏 ] ≅ 𝐶𝑜𝑛(𝐿), the lattice of all congruences of 𝐿. Definition 2.8 [6] Let 𝐼 = [𝑎, 𝑏] be an interval of a lattice 𝐿. The set 𝐼 × 𝐶2 is formed using the two-element chain 𝐶2 = {0,1}. The set 𝐿[𝐼] = (𝐿 ∖ 𝐼) ∪ (𝐼 × 𝐶2) is the lattice given by the ordering: for 𝑥, 𝑦 ∈ 𝐿[𝐼] and 𝑖, 𝑗 ∈ 𝐶2; 𝑥 ≤ 𝑦 if 𝑥 ≤ 𝑦 in 𝐿; (𝑥, 𝑖) ≤ 𝑦 if 𝑥 ≤ 𝑦 in 𝐿; 𝑥 ≤ (𝑦, 𝑗) if 𝑥 ≤ 𝑦 in 𝐿; (𝑥, 𝑖) ≤ (𝑦, 𝑗) if 𝑥 ≤ 𝑦 in 𝐿 and 𝑖 ≤ 𝑗 in 𝐶2. 𝐿[𝐼] is the lattice got by doubling of the interval 𝐼in𝐿. This is Day's definition of doubling of intervals. 2. Results and Discussions In this section, we examine whether [𝐶𝑊(𝐿𝑛)](𝐼) is 0 - distributive or not. It turns out that [𝐶𝑊(𝐿𝑛)](𝐼) remains 0-distributive in case of lower and upper intervals in 𝐶𝑊(𝐿𝑛), whereas in the case of an intermediate interval, 0 – distributivity gets affected. Theorem 3.1 If Ln is a chain of n elements, then [CW(Ln)](I) is 0 – distributive where I is a lower interval in CW(Ln). Proof. Let 𝐿𝑛 = { 0 ≺ 𝑥1 ≺ 𝑥2 ≺ 𝐼 ≺ 𝑥𝑛−1 = 1 } be a chain of 𝑛 elements. Let 𝐶𝑊(𝐿𝑛) be the lattice of all weak congruences of 𝐿𝑛. Let 𝐼 = [𝜙, 𝜃] where 𝜃 is a proper congruence relation of 𝐿𝑛. Let [𝐶𝑊(𝐿𝑛)](𝐼) be the doubling of 𝐶𝑊(𝐿𝑛) by the interval 𝐼. Let 𝐴, 𝐵, 𝐶 ∈ [𝐶𝑊(𝐿𝑛)](𝐼), where 𝐼 is a lower interval of 𝐶𝑊(𝐿𝑛) such that 𝐴 ∧ 𝐵 = (𝜙, 0) and 𝐴 ∧ 𝐶 = (𝜙, 0). To prove that, [𝐶𝑊(𝐿𝑛)](𝐼) is 0 – distributive. That is, we have to prove that 𝐴 ∧ (𝐵 ∨ 𝐶) = (𝜙, 0) (3.1) Suppose({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ (𝐵 ∨ 𝐶) ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴and ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 𝑎𝑛𝑑 ({(𝑥𝑖 , 𝑥𝑖 ) }, 0) ≤ 𝐵 or ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 or({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ both 𝐵 and 𝐶 or incomparable with both. Now,𝐴, 𝐵, 𝐶 ∈ [𝐶𝑊(𝐿𝑛)](𝐼) ⇒ 𝐴 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 or 𝐴 ∈ 𝐼 × 𝐶2, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 or 𝐵 ∈ 𝐼 × 𝐶2, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 or𝐶 ∈ 𝐼 × 𝐶2. The following cases arise: i.𝐴, 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 ii.𝐴, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐶 ∈ 𝐼 × 𝐶2 iii.𝐴 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐼 × 𝐶2 iv.𝐴, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐵 ∈ 𝐼 × 𝐶2 v.𝐴, 𝐵, 𝐶 ∈ 𝐼 × 𝐶2 vi.𝐴, 𝐵 ∈ 𝐼 × 𝐶2 𝑎𝑛𝑑 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 vii.𝐴, 𝐶 ∈ 𝐼 × 𝐶2 𝑎𝑛𝑑 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 viii.𝐴 ∈ 𝐼 × 𝐶2 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 Case (i): Let 𝐴, 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. 𝐴, 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 implies that 𝐴, 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛). Therefore, (3.1) follows as 𝐶𝑊(𝐿𝑛) is 0- distributive. 44 Gladys Mano Amirtha V & D. Premalatha Case (ii): Let 𝐴, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 and 𝐶 ∈ 𝐼 × 𝐶2, that is, 𝐶 = (𝜃3, 𝑗) where 𝑗 = either 0 or 1. (I) Let({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐵 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐵. This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (II) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 = (𝜃3, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3. Also, {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∧ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3 , 𝑗) = 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (III) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 and 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 and(𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃3 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐵 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐵. This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (IV) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. But, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = 𝐵 ∨ (𝜃3, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 ∨ 𝜃3 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃3. By symmetry and transitivity, (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 or(𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶.Again we have 𝐴 ∧ 𝐵 ≠ (𝜙, 0)or 𝐴 ∧ 𝐶 ≠ (𝜙, 0).This contradiction proves (3.1). Case (iii): Let 𝐴 ∈ 𝐶𝑊 (𝐿𝑛) ∖ 𝐼 and 𝐵, 𝐶 ∈ 𝐼 × 𝐶2.Let 𝐵 = (𝜃2, 𝑗) and 𝐶 = (𝜃3, 𝑗) where 𝑗 = either 0 or 1. (I) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 = (𝜃2, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2. Also, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃2 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃2, 𝑗) = 𝐴 ∧ 𝐵. This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (II) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 = (𝜃3, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3.Also, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (III) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 and 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2and {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃2 and{(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃2, 0) = 𝐴 ∧ 𝐵 and ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶 This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0) and 𝐴 ∧ 𝐶 = (𝜙, 0). (IV) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. But, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝜃2 ∨ 𝜃3, 𝑗) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 ∨ 𝜃3 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃3 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃3, by symmetry and transitivity in 𝜃2 and 𝜃3. ⇒ ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ (𝜃2, 0) 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ (𝜃3, 0). That is, ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ 𝐵 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ 𝐶. Therefore, we have ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶. This is a contradiction to our assumption. Therefore, (3.1) holds. Case (iv): Let 𝐴, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 and 𝐵 ∈ 𝐼 × 𝐶2, that is, 𝐵 = (𝜃2, 𝑗) where 𝑗 = either 0 or 1. (I) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 = (𝜃2, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2 . Also, {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃2 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃2, 𝑗) = 𝐴 ∧ 𝐵. This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (II) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (III) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 and 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2and (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). 45 On the effect of doubling of intervals on the 0-distributive property of the lattice of weak congruences of chains (IV) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. But, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝜃2 ∨ 𝐶, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 ∨ 𝐶 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐶. By symmetry and transitivity, (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶. This is a contradiction to our assumption. Therefore, (3.1) holds. Case (v): Let 𝐴, 𝐵, 𝐶 ∈ 𝐼 × 𝐶2, that is, 𝐴 = (𝜃1, 𝑗), 𝐵 = (𝜃2, 𝑗), 𝐶 = (𝜃3, 𝑗) where 𝑗 = either 0 or 1. (I) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 = (𝜃2, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2. Also, {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∧ 𝜃 2 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∧ 𝜃2, 𝑗) = 𝐴 ∧ 𝐵. This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (II) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 = (𝜃3, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3. Also,{(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃 3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (III) Let({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑎𝑛𝑑 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2 𝑎𝑛𝑑 {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃2and{(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃2, 0) = 𝐴 ∧ 𝐵 and ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0) 𝑎𝑛𝑑 𝐴 ∧ 𝐶 = (𝜙, 0). (IV) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. But, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝜃2 ∨ 𝜃3, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 ∨ 𝜃3 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that(𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ (𝜃2, 0) 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ (𝜃3, 0). That is, ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ 𝐵 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ 𝐶. By symmetry and transitivity in 𝐵 and 𝐶, we have ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶. This is a contradiction to our assumption. Therefore, (3.1) holds. Case (vi): Let 𝐴, 𝐵 ∈ 𝐼 × 𝐶2, that is, 𝐴 = (𝜃1, 𝑗), 𝐵 = (𝜃2, 𝑗) where j = either 0 or 1 and 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. (I) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 = (𝜃2, 𝑖) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2. Also,{(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∧ 𝜃2 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∧ 𝜃2, 𝑗) = 𝐴 ∧ 𝐵. This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (II) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (III) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑎𝑛𝑑 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 𝑎𝑛𝑑 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 ∩ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (IV) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. But, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝜃2 ∨ 𝐶, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 ∨ 𝐶 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐶. By symmetry and transitivity, we have (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶. This is a contradiction to our assumption.Therefore, (3.1) holds. Case (vii): Let 𝐴, 𝐶 ∈ 𝐼 × 𝐶2, that is, 𝐴 = (𝜃1, 𝑗), 𝐶 = (𝜃3, 𝑗) where 𝑗 = either 0 or 1 and 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. (I) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐵 46 Gladys Mano Amirtha V & D. Premalatha ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐵. This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (II) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 = (𝜃3, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐶. Also, {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∧ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶. This is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (III) Let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑎𝑛𝑑 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 𝑎𝑛𝑑 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃3 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐵 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐵.This is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (IV) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. But, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝐵 ∨ 𝜃3, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 ∨ 𝜃3. ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃3. By symmetry and transitivity, we have (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶.This is a contradiction to our assumption. Therefore, (3.1) holds. Case (viii): Let 𝐴 ∈ 𝐼 × 𝐶2, 𝐵 and 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. Let 𝐴 = (𝜃1, 𝑗)where 𝑗 = either 0 or 1. This case follows, since 𝐵 𝑎𝑛𝑑 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 implies 𝐵 𝑎𝑛𝑑 𝐶 ∈ 𝐶𝑊(𝐿𝑛) which is 0-distributive. Hence, [𝐶𝑊(𝐿𝑛)](𝐼) is 0-distributive whenever 𝐼 is a lower interval of 𝐶𝑊 (𝐿𝑛). Example 3.2 Consider the 0-distributive lattice 𝐶𝑊(𝐿4) where 𝐿4 is {0 ≺ 𝑎 ≺ 𝑏 ≺ 1}. Figure 1. 𝐶𝑊(𝐿4). Consider the interval 𝐼 = [𝜙, 𝑙32] in the above lattice Figure 1. Let 𝐶2 = {0,1} be the two-element chain. We can form the new lattice [𝐶𝑊(𝐿4)](𝐼) = {𝐶𝑊(𝐿4) ∖ 𝐼} ∪ (𝐼 × 𝐶2) given in Figure 2. 47 On the effect of doubling of intervals on the 0-distributive property of the lattice of weak congruences of chains Figure 2. [𝐶𝑊(𝐿4)](𝐼) where 𝐼 = [𝜙, 𝑙32]. Theorem 3.3 [𝐶𝑊(𝐿𝑛)](𝐼) is 0 - distributive, when 𝐼 is an upper interval of 𝐶𝑊(𝐿𝑛). Proof. Let 𝐼 = [{(1,1)}, 𝜏]. Let 𝐴, 𝐵, 𝐶 ∈ [𝐶𝑊(𝐿𝑛)](𝐼) such that 𝐴 ∧ 𝐵 = 𝜙, 𝐴 ∧ 𝐶 = 𝜙. Claim:𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙. (3.2) There can be two possibilities, that is, 𝐴 maybe in 𝐼 × 𝐶2 or 𝐴 maynot be in 𝐼 × 𝐶2. The following cases arise: i.𝐴 ∈ 𝐼 × 𝐶2 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 ii.𝐴 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 iii.𝐴 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐼 × 𝐶2 iv.𝐴, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐶 ∈ 𝐼 × 𝐶2 Case (i): Let 𝐴 ∈ 𝐼 × 𝐶2 and 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. We note that either 𝐴 ∧ (𝐵 ∨ 𝐶) ∈ 𝐼 × 𝐶2 or 𝐴 ∧ (𝐵 ∨ 𝐶) ∉ 𝐼 × 𝐶2. Suppose 𝐴 ∧ (𝐵 ∨ 𝐶) ≠ 𝜙. Therefore, there exists {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 ∧ (𝐵 ∨ 𝐶) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 and{(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 ∨ 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 and {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 or {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐶 or{(𝑥𝑖 , 𝑥𝑖 )} ≤ both 𝐵 and 𝐶 or incomparable with both. ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 𝑎𝑛𝑑 {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 which is a contradiction, since 𝐴 ∧ 𝐵 ≠ 𝜙. Similarly, we get a contradiction, when {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 𝑎𝑛𝑑 {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐶 and when {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 and {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 & 𝐶.So, 𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙. When {(𝑥𝑖 , 𝑥𝑖 )} ≰ 𝑏𝑜𝑡ℎ 𝐵 & 𝐶, then there exists 𝑥𝑘 such that {(𝑥𝑖 , 𝑥𝑘 )} ≤ 𝐵 or {(𝑥𝑖 , 𝑥𝑘 )} ≤ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐶. So, by symmetry and transitivity, we have (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵.So, 𝐴 ∧ 𝐵 ≥ {(𝑥𝑖 , 𝑥𝑖 )}, a contradiction again. So, (3.2) holds. Case (ii): Let 𝐴 ∈ 𝐶𝑊 (𝐿𝑛) ∖ 𝐼 and 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. As 𝐶𝑊(𝐿𝑛) is 0 - distributive, it follows that 𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙. Case (iii): Let 𝐴 ∈ 𝐶𝑊 (𝐿𝑛) ∖ 𝐼 and 𝐵, 𝐶 ∈ 𝐼 × 𝐶2. Suppose 𝐴 ∧ (𝐵 ∨ 𝐶) ≠ 𝜙. 48 Gladys Mano Amirtha V & D. Premalatha Therefore, there exists (𝑥𝑖 , 𝑥𝑖 ) ≤ 𝐴 ∧ (𝐵 ∨ 𝐶).As in Case (i),𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙 follows. Case (iv): Let 𝐴, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 and 𝐶 ∈ 𝐼 × 𝐶2. Suppose 𝐴 ∧ (𝐵 ∨ 𝐶) ≠ 𝜙. Therefore, there exists (𝑥𝑖 , 𝑥𝑖 ) ≤ 𝐴 ∧ (𝐵 ∨ 𝐶). As in Case (i),𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙 follows. Hence, for an upper interval 𝐼, [𝐶𝑊(𝐿𝑛)](𝐼) is 0-distributive. Example 3.4 Consider the interval [𝑙1, 𝜏] in Figure 1. The lattice formed by the doubling of the interval is given in Figure 3. Figure 3. [𝐶𝑊(𝐿4)](𝐼) where 𝐼 = [𝑙1, 𝜏]. Remark 3.5 The property of 0 - distributivity doesn’t hold if we consider an intermediate interval I of [𝐶𝑊(𝐿𝑛)]. Example 3.6 Consider the intermediate interval 𝐼 = [𝑙4, 𝑙20] in Figure 1. Consider the elements (𝑙4, 1) and (𝑙10, 0) disjoint with 𝑙3, that is, 𝑙3 ∧ (𝑙4, 1) = 𝜙 and 𝑙3 ∧ (𝑙10, 0) = 𝜙. Now, 𝑙3 ∧ [(𝑙4, 1) ∨ (𝑙10, 0)] = 𝑙3 ∧ (𝑙10, 1) = 𝑙3 ≠ 𝜙. Figure 4. [𝐶𝑊(𝐿4)[𝐼] where 𝐼 = [𝑙4, 𝑙20]. 49 On the effect of doubling of intervals on the 0-distributive property of the lattice of weak congruences of chains References [1] Balasubramani, P., Stone Topologies of the set of prime filters of a 0- distributive lattice, Indian Journal of Pure and Applied math., 35(2) (2004), 149 - 158. [2] Chajda.I., Congruence distributivity in varieties with constant, Archivum Mathematicum., 22 (1983), 121 - 124. [3] A, Day, A simple solution of the word problem for lattice, Canad. Math. Bull. 13 (1970), 253-254 [4] A. Day, Herb Gaskill and Werner Poguntke, Distributive lattice with finite projective covers, Pacific Journal of Math., 81 (1979). [5] A. Day, Doubling constructions in Lattie theory, Can. J. 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