Generalized Almost Distributive Lattices-I

Southeast Asian Bulletin of Mathematics (2009) 33: 1175–1188

Southeast Asian Bulletin of Mathematics c SEAMS. 2009 °

Generalized Almost Distributive Lattices-I G.C. Rao1 , Ravi Kumar Bandaru2 and N. Rafi3 Department of Mathematics, Andhra University, Visakhapatnam, Andhra Pradesh 530003, India E-mail: [email protected] , [email protected] and [email protected] Received 3 April 2009 Accepted 18 June 2009 Communicated by K.P. Shum

AMS Mathematics Subject Classification (2000): 06D99, 06D15 Abstract. The concept of a GADL as a generalization of an ADL is introduced. Necessary and sufficient conditions for a GADL to become a distributive lattice and a GADL to become an ADL are obtained. We also study the maximal sets in a GADL and give equivalent conditions for a GADL to become a distributive lattice in terms of maximal sets. Keywords: Almost distributive lattice(ADL); Generalized almost distributive lattice (GADL); Maximal set; Amicable set.

1. Introduction The concept of an Almost Distributive Lattice(ADL) was introduced by Swamy and Rao [5] as a common abstraction of all existing ring theoretic generalizations of a Boolean algebra (like regular rings, p−rings, biregular rings, associate rings, P1 −rings etc.) on one hand and distributive lattices on the other. The class of ADLs inherit almost all the properties of a distributive lattice except possibly the commutativity of ∧, ∨ and the right distributivity of ∨ over ∧. Later different classes of ADLs like Pseudo-complementation on ADLs [4], Stone ADLs [6] and Normal ADLs [7] were characterized. In this paper, we introduce the class of a GADL as a generalization of an ADL. This class of GADLs includes the class of ADLs properly and retain many important properties of ADLs. In Section 3, we give a number of equivalent conditions for a GADL to become an ADL as well

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G.C. Rao, R.K. Bandaru and N. Rafi

as to become a distributive lattice. In Section 4, we study the maximal sets in a GADL and give the necessary and sufficient conditions for a GADL to become a distributive lattice in terms of maximal sets.

2. Preliminaries Definition 2.1. [3] An Almost Distributive Lattice (ADL) is an algebra (L, ∨, ∧) of type (2, 2) satisfying the following axioms: 1) 2) 3) 4) 5)

(x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z), x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), (x ∨ y) ∧ y = y, (x ∨ y) ∧ x = x, x ∨ (x ∧ y) = x.

It can be seen directly that every distributive lattice is an ADL. If there is an element 0 ∈ L such that 0 ∧ a = 0 for all a ∈ L, then (L, ∨, ∧, 0) is called an ADL with 0. An ADL (L, ∨, ∧, 0) satisfies many properties satisfied by a distributive lattice with 0. Lemma 2.2. [3] Let (L, ∨, ∧, 0) be an ADL with 0. Then it satisfies the following: 1) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c), 2) a ∨ 0 = a for all a, b, c ∈ L. Lemma 2.3. [5] If (L, ∨, ∧, 0) is an ADL, for any a, b ∈ L, define a ≤ b if and only if a = a ∧ b or equivalently, a ∨ b = b, then ≤ is a partial ordering on L and for any a, b, c ∈ L, we have the following: (1) a ∨ b = a ⇔ a ∧ b = b, (2) a ∨ b = b ⇔ a ∧ b = a, (3) a ∨ b = b ∨ a whenever a ≤ b, (4) ∧ is associative in L, (5) a ∧ b ∧ c = b ∧ a ∧ c, (6) (a ∨ b) ∧ c = (b ∨ a) ∧ c, (7) a ∧ b = 0 ⇔ b ∧ a = 0, (8) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c), (9) a ∧ (a ∨ b) = a, (a ∧ b) ∨ b = b, and a ∨ (b ∧ a) = a, (10) a ≤ a ∨ b and a ∧ b ≤ b, (11) a ∧ a = a and a ∨ a = a, (12) 0 ∨ a = a and a ∧ 0 = 0, (13) If a ≤ c, b ≤ c then a ∧ b = b ∧ a and a ∨ b = b ∨ a, (14) a ∨ b = (a ∨ b) ∨ a.

Generalized Almost Distributive Lattices-I

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As usual, an element m ∈ L is called maximal if it is a maximal element in the partially ordered set (L, ≤). That is, for any a ∈ L, m ≤ a ⇒ m = a. Theorem 2.4. [5] Let L be an ADL and m ∈ L. Then the following conditions are equivalent. 1) m is maximal with respect to ≤, 2) m ∨ a = m, for all a ∈ L, 3) m ∧ a = a, for all a ∈ L.

3. Generalized almost Distributive Lattice In this section, we introduce the concept of a Generalized Almost Distributive Lattice. We give below a few examples. Definition 3.1. An algebra (L, ∨, ∧) of type (2, 2) is called a Generalized Almost Distributive Lattice if it satisfies the following axioms: ( As ∧) (x ∧ y) ∧ z = x ∧ (y ∧ z), ( LD ∧) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), ( LD ∨) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z), ( A1 ) x ∧ (x ∨ y) = x, ( A2 ) (x ∨ y) ∧ x = x, ( A3 ) (x ∧ y) ∨ y = y. Example 3.2. Let R = {a, b, c}. Define two binary operations ∨ and ∧ on R as follows: ∨ a b c

a a b c

b b b c

c a b c

∧ a b c

a a a a

b a b a

c c c c

Hence the algebra (R, ∨, ∧) is a Generalized Almost Distributive Lattice. Example 3.3. Let R = {a, b, c}. Define two binary operations ∨ and ∧ on R as follows: ∨ a b c

a a a c

b a b c

c a b c

∧ a b c

a a b b

b b b b

c c c c

Hence the algebra (R, ∨, ∧) is a Generalized Almost Distributive Lattice. For brevity, we will refer to this Generalized Almost Distributive Lattice as GADL. The GADL (R, ∨, ∧) in Example 3.2 is not an ADL for (c ∨ b) ∧ b 6= b.

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In the following L stands for a GADL (L, ∨, ∧). Lemma 3.4. For any a, b ∈ L, we have (I∨) a ∨ a = a, (I∧) a ∧ a = a, ( A4 ) a ∨ (a ∧ b) = a, ( A5 ) a ∨ (b ∧ a) = a. Proof. Let a, b ∈ L. Then (I∨) a ∨ a = ((a ∨ a) ∧ a) ∨ a (∵ (a ∨ b) ∧ a = a) = a ( by A3 ). (I∧) a ∧ a = a ∧ (a ∨ a) ( by (I∨)) = a ( by A1 ). (A4 ) a ∨ (a ∧ b) = (a ∨ a) ∧ (a ∨ b) ( by LD ∨) = a ∧ (a ∨ b) ( by I∨) =a ( by A1 ). Also (A5 ) a ∨ (b ∧ a) = (a ∨ b) ∧ (a ∨ a) by LD ∨) = (a ∨ b) ∧ a ( by I∨) =a ( by A2 ).

Lemma 3.5. For any a, b ∈ L, we have (1) a ∧ b = b ⇒ a ∨ b = a, (2) a ∨ b = b ⇔ a ∧ b = a. Proof. Let a, b ∈ L. (1) Suppose a ∧ b = b. Then a ∨ b = a ∨ (a ∧ b) = a by A4 . (2) Suppose a ∨ b = b. Then a ∧ b = a ∧ (a ∨ b) = a by A1 . Conversely, if a ∧ b = a, then, by A3 , we get b = (a ∧ b) ∨ b = a ∨ b. In a GADL, the converse of Lemma 3.5(1) does not hold. In Example 3.2 we observe that c ∨ b = c but c ∧ b = a. Using the absorption laws A1 through A5 , from the above Lemma, we have the following: Lemma 3.6. For any a, b ∈ L, we have (1) a ∨ (a ∨ b) = a ∨ b, (2) b ∧ (a ∧ b) = a ∧ b, (3) a ∧ (b ∧ a) = b ∧ a. In view of Lemma 3.5, we introduce a partial ordering on L. For any a, b ∈ L we say that a is less than or equal to b and write a ≤ b, if a ∧ b = a or, equivalently, a ∨ b = b. Then it can be easily verified that ≤ is a partial ordering on L. Regarding the remaining absorption laws we have the following theorem:


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Theorem 3.7. For any a, b ∈ L, the following are equivalent: (1) (a ∧ b) ∨ a = a, (2) a ∧ (b ∨ a) = a, (3) (b ∧ a) ∨ b = b, (4) b ∧ (a ∨ b) = b, (5) a ∧ b = b ∧ a, (6) a ∨ b = b ∨ a. Proof. Let a, b ∈ L. By LD ∧ and I∧, we have a ∧ (b ∨ a) = (a ∧ b) ∨ (a ∧ a) = (a ∧ b) ∨ a so that conditions (1) and (2) are equivalent. By interchanging the roles of a and b, we get the equivalence of (3) and (4). Now, we prove the equivalence of (5) and (1). Assume (5). Then (a ∧ b) ∨ a = (b ∧ a) ∨ a = a (by A3 ). Thus we have (5) ⇒ (1). Now, assume (1). Then b∧a = = = = =

b ∧ {(a ∧ b) ∨ a} = {b ∧ (a ∧ b)} ∨ (b ∧ a) (a ∧ b) ∨ (b ∧ a) by Lemma 3.6 (a ∧ b) ∨ {a ∧ (b ∧ a)} by Lemma 3.6 a ∧ {b ∨ (b ∧ a)} by (LD∧) a ∧ b.

Thus (1) ⇒ (5). Now we prove (1) ⇒ (3). Assume (1). Then a ∧ b = b ∧ a. Now (b ∧ a) ∨ b = (a ∧ b) ∨ b = b. Now we prove (3) implies (5). Assume (3). Then a∧b = = = = = =

a ∧ {(b ∧ a) ∨ b} {a ∧ (b ∧ a)} ∨ (a ∧ b) by LD ∧ (b ∧ a) ∨ (a ∧ b) by Lemma 3.6 (b ∧ a) ∨ {b ∧ (a ∧ b)} by Lemma 3.6 b ∧ {a ∨ (a ∧ b)} by LD ∧ b ∧ a by A4 .

Therefore a ∧ b = b ∧ a. We complete the proof by proving the equivalence of (6) and (2). Assume (6). Then a ∧ (b ∨ a) = a ∧ (a ∨ b) = a. Finally, assume (2). Then (b ∧ a) ∨ b = b by the equivalence of (2) and (3) and hence b ∧ (a ∨ b) = b. Now, b∨a = = = = = =

b ∨ {(a ∨ b) ∧ a} by A2 {b ∨ (a ∨ b)} ∧ (b ∨ a) by LD ∨ (a ∨ b) ∧ (b ∨ a) by Lemma 3.5(2) (a ∨ b) ∧ {a ∨ (b ∨ a)} by our assumption and Lemma 3.5(2) a ∨ {b ∧ (b ∨ a)} by LD ∨ a ∨ b by A1 .

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Therefore a ∨ b = b ∨ a. Hence the theorem is proved. Now we give some more equivalent conditions in terms of the partial ordering ‘ ≤’ that are equivalent to commutativity of ∨, that is, a ∨ b = b ∨ a. Theorem 3.8. For any a, b ∈ L, the following are equivalent: (1) (2) (3) (4) (5) (6)

a ∨ b = b ∨ a, b ≤ a ∨ b, There exists c ∈ L such that a ≤ c and b ≤ c, a ∨ b is the lub of a and b in (L, ≤), a ≤ b ∨ a, b ∨ a is the lub of a and b in (L, ≤).

Proof. Let a, b ∈ L. Since a ≤ a ∨ b and b ≤ b ∨ a, we have (1) ⇒ (2) ⇒ (3). Now assume (3). Then b ∧ (a ∨ b) = (b ∧ a) ∨ b = (b ∧ a) ∨ (b ∧ c) = b ∧ (a ∨ c) = b ∧ c = b. So b ≤ a ∨ b. Thus a ∨ b is an upper bound of a and b. If x ∈ L such that a ≤ x and b ≤ x, then (a ∨ b) ∧ x = (a ∨ b) ∧ (a ∨ x) = a ∨ (b ∧ x) = a ∨ b. So a ∨ b ≤ x and hence a ∨ b is the lub of a and b in L. Thus (3) ⇒ (4). Now assume (4). Then b ≤ a ∨ b, so that b ∧ (a ∨ b) = b and hence, by the equivalence of (4) and (6) in the above theorem, we get a ∨ b = b ∨ a. Thus the conditions (1) through (4) are equivalent. Now we prove the equivalence of (1) and (5). Since a ≤ a ∨ b, we get (1) ⇒ (5). Assume (5). Then a ∧ (b ∨ a) = a and hence, by the equivalence of (1) and (6) in the above Theorem, we have a ∨ b = b ∨ a. We complete the proof by proving the equivalence of (5) and (6). Assume (5). We know that b ≤ b ∨ a. Therefore b ∨ a is an upper bound of a and b. If x ∈ L such that a ≤ x and b ≤ x then (b ∨ a) ∧ x = (b ∨ a) ∧ (b ∨ x) = b ∨ (a ∧ x) = b ∨ a and hence b ∨ a is the lub of a and b in L. (6) ⇒ (5) is trivial. Hence the theorem. We give another set of equivalent conditions in the following theorem that are equivalent to commutativity of ∧. That is, a ∧ b = b ∧ a. Theorem 3.9. For any a, b ∈ L, the following are equivalent: (1) a ∧ b = b ∧ a, (2) a ∧ b ≤ a, (3) a ∧ b is the glb of a and b in (L, ≤),


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(4) b ∧ a ≤ b, (5) b ∧ a is the glb of a and b in (L, ≤). The following lemma is very useful in the GADL. Lemma 3.10. For any a, b, c ∈ L,

a ∧ b ∧ c = b ∧ a ∧ c.

Proof. Since a ∧ c ≤ c and b ∧ c ≤ c, by Theorem 3.8, we have (a ∧ c) ∨ (b ∧ c) = (b ∧ c) ∨ (a ∧ c). Hence by Theorem 3.7, we have (a ∧ c) ∧ (b ∧ c) = (b ∧ c) ∧ (a ∧ c) ⇒ a ∧ c ∧ b ∧ c = b ∧ c ∧ a ∧ c. Hence by Lemma 3.6(3), a ∧ b ∧ c = b ∧ a ∧ c. Theorem 3.11. Let (L, ∨, ∧) be a GADL. Then the following are equivalent: (1) (2) (3) (4) (5) (6) (7)

(L, ∨, ∧) is a distributive lattice, The poset (L, ≤) is directed above, a ∧ (b ∨ a) = a for all a, b ∈ L, The operation ∨ is commutative in L, The operation ∧ is commutative in L, The relation θ = {(a, b) ∈ L × L | a ∧ b = b} is anti-symmetric, The relation θ defined in (6) is a partial ordering on L.

Proof. The equivalence of (2), (3), (4) and (5) follows from Theorems 3.7 and 3.8. Since ≤ is a partial ordering, we have (5) ⇒ (6). Since the relation θ defined in (6) is always reflexive and transitive, we have (6) ⇔ (7). Since (1) ⇒ (2) is clear, we complete the Theorem by proving (7) ⇒ (1). Assume (7) and suppose a, b ∈ L. Then, by Lemma 3.10, (a∧b)∧(b∧a) = b∧a and (b∧a)∧(a∧b) = a∧b. So that the elements (a∧b, b∧a) and (b∧a, a∧b) belong to θ and hence a∧b = b∧a. Now, by Theorems 3.7, 3.8 and 3.9 we get that a ∧ b is the glb of a and b and a ∨ b is the lub of a and b and hence L is a lattice. Therefore, L is a distributive lattice. Corollary 3.12. For any a ∈ L, the set La = {x ∧ a | x ∈ L} = {x ∈ L | x ≤ a} is a distributive lattice under the induced operations ∨ and ∧, with a as its greatest element. Now we give the equivalent conditions for a GADL to become an ADL. Theorem 3.13. Let (L, ∨, ∧) be a GADL. Then the following are equivalent. (1) (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c) for all a, b, c ∈ L, (2) (a ∨ b) ∧ b = b for all a, b ∈ L, (3) (a ∨ b) ∧ c = (b ∨ a) ∧ c for all a, b, c ∈ L.

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Proof. (1) ⇒ (2) : Assume (1). Let a, b ∈ L. Then (a ∨ b) ∧ b = (a ∧ b) ∨ (b ∧ b) = (a ∧ b) ∨ b = b. (2) ⇒ (3) : Assume (2). i.e., (a ∨ b) ∧ b = b for all a, b ∈ L. We know that (a ∨ b) ∧ c ≤ c, (b ∨ a) ∧ c ≤ c. Then £ ¤ £ ¤ £ ¤ £ ¤ (a ∨ b) ∧ c ∧ (b ∨ a) ∧ c = (b ∨ a) ∧ c ∧ (a ∨ b) ∧ c £ ¤ £ ¤ ⇒(a ∨ b) ∧ c ∧ (b ∨ a) ∧ c = (b ∨ a) ∧ c ∧ (a ∨ b) ∧ c ⇒(a ∨ b) ∧ (b ∨ a) ∧ c = (b ∨ a) ∧ (a ∨ b) ∧ c h£ h£ ¤ £ ¤i ¤ £ ¤i ⇒ (a ∨ b) ∧ b ∨ (a ∨ b) ∧ a ∧ c = (b ∨ a) ∧ a ∨ (b ∨ a) ∧ b ∧ c ⇒(b ∨ a) ∧ c = (a ∨ b) ∧ c. (3) ⇒ (1) : Assume (3)£ and a, b ∈¤L. £Then ¤ (a ∧ c) ∨ (b ∧ c) = £ (a ∧ c) ∨ b¤ ∧ (a ∧ c) ∨ c = £(a ∧ c) ∨ b¤ ∧ c = £b ∨ (a ∧ c) ∧ c ¤ by (3) = h(b ∨ a) ∧ (b ∨ c) ∧ c £ ¤ £ ¤i = (b ∨ a) ∧ b ∨ (b ∨ a) ∧ c ∧ c h i = b ∨ [(b ∨ a) ∧ c] ∧ c £ ¤ = £b ∨ (b ∨ a)¤ ∧ (b ∨ c) ∧ c = b ∨ (b ∨ a) ∧ (c ∨ b) ∧ c by (3) = (b ∨ a) ∧ c = (a ∨ b) ∧ c. Therefore (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c).

Definition 3.14. Let (L, ∨, ∧) be a GADL. An element 0 ∈ L is called a zero element of L if (01 ) 0 ∧ a = 0 for all a ∈ L. We always denote the zero element of L, if it exists, by ‘0’. If L has 0, then the algebra (L, ∨, ∧, 0) is called a GADL with 0. Now we have the following Lemma 3.15. Let (L, ∨, ∧, 0) be a GADL with 0. Then, for any a ∈ L, the following hold: (02 ) a ∨ 0 = a, (03 ) 0 ∨ a = a, (04 ) a ∧ 0 = 0. The following lemma is immediate from Lemma 3.10.


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Lemma 3.16. Let (L, ∨, ∧, 0) be a GADL with 0. For any a, b ∈ L, a ∧ b = 0 if and only if b ∧ a = 0. Lemma 3.17. For any a, b ∈ L, a ∧ b = 0 ⇒ a ∨ b = b ∨ a. Proof. Follows from Theorem 3.7 Remark 3.18. If a ≤ b and x ∈ L, then we get that a ∧ x ≤ b ∧ x, x ∧ a ≤ x ∧ b and x ∨ a ≤ x ∨ b. But it is not true, in general, that a ≤ b need to imply that a ∨ x ≤ b ∨ x; in fact, L is a distributive lattice with 0 if and only if for any a, b ∈ L, a ≤ b implies that a ∨ x ≤ b ∨ x for all x ∈ L.

4. Maximal Sets Definition 4.1. For any a, b ∈ L, we say that a is compatible with b and write a ∼ b if a ∧ b = b ∧ a or, equivalently, a ∨ b = b ∨ a. A subset S of L is said to be compatible if a ∼ b for all a, b ∈ S. By a maximal set we mean a maximal compatible set. The following theorem can be directly verified using the above definition. Theorem 4.2. For any a, b ∈ L, the following hold: (1) a ∼ b ⇔ a ∧ b ∼ b ∧ a ⇔ a ∨ b ∼ b ∨ a, (2) a ∼ b ⇔ a ≤ x and b ≤ x for some x ∈ L, (3) a ≤ b ⇒ a ∼ b, (4) a ∼ b ⇔ a ∨ x = b ∨ x for some x ∈ L, (5) a ∼ b, c ∼ d ⇒ a ∧ c ∼ b ∧ d, (6) For any x, y ∈ L; x ∼ y ⇒ x ∧ y ∧ (c ∨ x) = x ∧ y ∧ (c ∨ y) for all c ∈ L, (7) For any x, y ∈ L; x ∼ y ⇒ x ∧ (c ∨ x) ∼ y ∧ (c ∨ y) for all c ∈ L, (8) a ∧ c ∼ b ∧ c, (9) a ∼ b ⇒ c ∧ a ∼ c ∧ b and c ∨ a ∼ c ∨ b, (10) a ∧ b ∼ a ⇔ L is distributive lattice, (11) a ∼ b, c ∼ b ⇒ a ∨ c ∼ b and a ∧ c ∼ b, (12) a ∼ b ⇒ (a ∨ c) ∧ b = (a ∧ b) ∨ (c ∧ b) for all c ∈ L, (13) The operation ∨ is right distributive over the operation ∧ in L if and only if (L, ∨, ∧, 0) is a distributive lattice with 0. Lemma 4.3. Let M be a maximal set in L and x ∈ L be such that x ∼ a for all a ∈ M. Then x ∈ M. Proof. Since a ∼ x for all a ∈ M and M is a compatible set, we get M ∪ {x} is a compatible set and hence, by the maximality of M, x ∈ M.

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Theorem 4.4. If M is a maximal set in L then M is a distributive lattice under the induced operations ∨ and ∧. Proof. Let M be a maximal set in L and a, b ∈ M. Then for any c ∈ M, we have a ∼ c, b ∼ c and hence a ∧ b ∼ c as well as a ∨ b ∼ c by Lemma 4.2(11). Therefore a ∧ b, a ∨ b are in M. Thus M is a distributive lattice under the induced operations ∨ and ∧ in L. We immediately have the following: Corollary 4.5. The following are equivalent: (1) L is a distributive lattice, (2) L is a compatible set, (3) L is a maximal set. Proposition 4.6. Let M be a maximal set in L and a ∈ M. Then, for any x ∈ L, x ∧ a ∈ M. Proof. Let x ∈ L. Then, for any b ∈ M, x ∧ a ∧ b = x ∧ b ∧ a = b ∧ x ∧ a. So that x ∧ a ∼ b. Hence by Lemma 4.3, x ∧ a ∈ M. Corollary 4.7. Let M be a maximal set. Then M is an initial segment in the poset (L, ≤). That is, for any x ∈ L and a ∈ M, x ≤ a implies x ∈ M. Since ∼ is reflexive, by Zorn’s lemma, for any a ∈ L, there exists a maximal set containing a and hence, every GADL is a set union of its maximal sets. Definition 4.8. Let M be a maximal set in L. Then an element x ∈ L is said to be M −amicable if there exists d ∈ M such that d ∧ x = x. If every element of L is M −amicable, then we call M an amicable set. Lemma 4.9. Let M be a maximal set and x ∈ L be M −amicable. Then there exists an element a ∈ M with the following properties: (1) a ∧ x = x, (2) b ∈ L, b ∧ x = x ⇒ b ∧ a = a. Proof. Since x is M −amicable, there exists c ∈ M such that c ∧ x = x. So a = x ∧ c ∈ M. Now a ∧ x = x ∧ c ∧ x = c ∧ x = x and if b ∈ L and b ∧ x = x, then b ∧ a = b ∧ x ∧ c = x ∧ c = a. Hence the lemma follows. If b ∈ M is in the above Lemma 4.9, then a = b ∧ a = a ∧ b so that a ≤ b. Corollary 4.10. Let M be a maximal set and x ∈ L an M −amicable element. Then there exists a unique element a ∈ M such that a ∧ x = x and x ∧ a = a.


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We denote the element a of M in the above corollary by xM and it is the smallest element of M with the property xM ∧ x = x. Observe that xM depends on M as well as on x and if x ∈ M then x = xM . Now we prove the following: Lemma 4.11. Let M be a maximal set, x ∈ L be M −amicable and a ∈ L such that x ∧ a = a. Then a is M −amicable and aM ≤ xM . Proof. Since xM ∧a = xM ∧x∧a = x∧a = a, a is M −amicable. Also aM ≤ xM . Corollary 4.12. Let M be a maximal set and x ∈ M be M −amicable. Then xM is the largest element of M with the property x ∧ xM = xM and xM is the smallest element of M with the property xM ∧ x = x. Corollary 4.13. Let M be a maximal set and x ∈ L be M −amicable. Then, for any a ∈ L, a∧x = x and x∧a = a if and only if a is M −amicable and xM = aM . In an ADL the associativity of the operation ∨ is not known. But in a GADL the associativity of the operation ∨ does not hold. For, from Example 3.2, we observe that a ∨ (c ∨ b) = a ∨ c = a 6= b = a ∨ b = (a ∨ c) ∨ b, ∨ is not associative. Definition 4.14. A GADL (L, ∨, ∧) is said to be associative if the operation ∨ in L is associative. If M is a maximal set in L, then we denote the set of all M − amicable elements of L by AM (L). Now we prove that AM (L) is a GADL with the induced operations. Theorem 4.15. Let M be a maximal set in L. Then (AM (L), ∨, ∧) is a GADL. Moreover, if L is associative then for any x, y ∈ L, we have (x ∨ y)M = xM ∨ y M and (x ∧ y)M = xM ∧ y M . Proof. Let x, y ∈ AM (L). Then there exists a unique element xM ∈ M such that xM ∧ x = x, x ∧ xM = xM and there exists a unique element y M ∈ M such that y M ∧ y = y, y ∧ y M = y M . Now, (xM ∨ y M ) ∧ (x ∨ y) = = = = =

{(xM ∨ y M ) ∧ x} ∨ {(xM ∨ y M ) ∧ y} {(xM ∨ y M ) ∧ xM ∧ x} ∨ {(xM ∨ y M ) ∧ y M ∧ y} (xM ∧ x) ∧ {(y M ∨ xM ) ∧ y M ∧ y} x ∨ (y M ∧ y) x ∨ y.

Therefore x∨y ∈ AM (L). Hence by Corollary 4.10, (x∨y)M = (x∨y)∧(xM ∨y M ). Now, xM ∧y M ∧x∧y = xM ∧x∧y M ∧y = x∧y. Also we get that x∧y∧xM ∧y M = xM ∧y M . Therefore x∧y ∈ AM (L) and (x∧y)M = xM ∧y M . Hence (AM (L), ∨, ∧) is a GADL. Let L be associative GADL and x, y ∈ AM (L). Then

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(x ∨ y) ∧ (xM ∨ y M ) = = = = = = = = = Therefore, by Corollary


{(x ∨ y) ∧ xM } ∨ {(x ∨ y) ∧ y M } {(x ∨ y) ∧ x ∧ xM } ∨ {(x ∨ y) ∧ y M } (x ∧ xM ) ∨ {(x ∨ y) ∧ y M } xM ∨ {(x ∨ y) ∧ y M } {xM ∨ (x ∨ y)} ∧ (xM ∨ y M ) {(xM ∨ x) ∨ y} ∧ (xM ∨ y M ) (xM ∨ y) ∧ (xM ∨ y M ) xM ∨ (y ∧ y M ) xM ∨ y M . 4.10, we get that (x ∨ y)M = xM ∨ y M .

Proposition 4.16. Let M be a maximal set, x, y ∈ L be M −amicable and x ∼ y. Then xM = y M if and only if x = y. Proof. Suppose xM = y M . Then x = xM ∧x = y M ∧x = y∧y M ∧x = y M ∧y∧x = y ∧ x = x ∧ y so that x ≤ y. Hence by symmetry, x = y. Corollary 4.17. Let M be an amicable set. If x, y ∈ M then xM = y M ⇒ x = y. For any maximal set M of L, clearly M ⊆ AM (L) ⊆ L. Now we prove the following result. Theorem 4.18. Let M be a maximal set in L. Then the following are equivalent: (1) M = AM (L), (2) M = L, (3) (L, ∨, ∧) is a distributive lattice. Proof. (1) ⇒ (2) : Assume that (1) holds. Let x ∈ L. Then, for any a ∈ M, we have a ∧ x ∈ M. Also x ∧ a ∈ M. Therefore a ∧ x ∼ x ∧ a and hence x ∼ a. Hence x ∈ M. Thus M = L. (2) ⇒ (3) : This part follows from Theorem 4.4. (3) ⇒ (1) : Assume that (3) holds. If L is a lattice, then L is the only maximal set in L. Hence M = L. Thus M = AM (L). If M is an amicable set, then we know that each x ∈ L is M −amicable and M is an initial segment. Now we prove the following theorem: Theorem 4.19. Let M be a compatible set in L satisfying the following (1) for each x ∈ L, there exists d ∈ M such that d ∧ x = x, (2) M is an initial segment in (L, ≤). Then M is maximal and hence an amicable set. Proof. Let y ∈ L such that y ∼ a for all a ∈ M. By (1), there exists d ∈ M such that d ∧ y = y. Since y ∼ d, we have y = d ∧ y = y ∧ d and hence y ≤ d. Therefore


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y ∈ M (by (2)). Therefore M is a maximal set and hence an amicable set. Theorem 4.20. Let L be an associative GADL. Let M be a maximal set and 0 M 0 an amicable set in L. Then the mapping a 7→ aM is an isomorphism of the distributive lattice (M, ∨, ∧) into the distributive lattice (M 0 , ∨, ∧). Further if M is also amicable then the above mapping is a surjection. 0

Proof. Define f : M → M 0 by f (a) = aM for all a ∈ M. Then we get that f is an isomorphism of M into M 0 by Theorem 4.15 and Corollary 4.17. Suppose M is also amicable. Let x ∈ M 0 . Then x is M −amicable and xM ∧x = x, x∧xM = xM . 0 Since x ∈ M 0 , we have (xM )M ≤ x by Corollary 4.12. Now, x = xM ∧ x = 0 0 0 (xM )M ∧ xM ∧ x = (xM )M ∧ x = (xM )M . Therefore f is a surjection. Definition 4.21. Let M be a maximal set in L. An element u of L is said to be a unielement of M if a ≤ u for all a ∈ M. Observe that a unielement of a maximal set, if it exists, is unique and is in M. As usual, we say that an element x ∈ L is maximal if no element of L dominates x. We immediately have the following: Proposition 4.22. For any m ∈ L, m is maximal if and only if m ∨ x = m for all x ∈ L. Definition 4.23. Let (L, ∨, ∧) be a GADL. An element e ∈ L is said to be left identity element in L if e ∧ x = x for all x ∈ L. Note that every left identity element is maximal element but converse need not be true. In Example 3.2, we observe that c is maximal but not left identity element. Proposition 4.24. Let (L, ∨, ∧) be an associative GADL. If L has a maximal element n then, for each x ∈ L, there exists a maximal element m ∈ L such that x ≤ m. Proof. Follows from the fact that, for any x ∈ L, n ∨ x is also a maximal element of L. Theorem 4.25. Suppose L has left identity element. Then every amicable set has a unielement. Proof. Suppose M is an amicable set in L. Let e be a left identity element of L. Then e is M −amicable. Let x ∈ M. Then x = e∧x = eM ∧e∧x = eM ∧x = x∧eM since x, eM ∈ M. Therefore x ≤ eM . Hence eM is a unielement of M.

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Lemma 4.26. Let M be a maximal set in L with unielement u. Then u is a maximal element of L and M = {x ∧ u | x ∈ L}. Proof. Let x ∈ L. Then, for any a ∈ M, a ≤ u ≤ u ∨ x, so a ∼ u ∨ x and hence u ∨ x ∈ M. Therefore, u = u ∨ x. Hence by Proposition 4.22, u is a maximal element of L and the rest is clear. The converse of the above Lemma is also true. It is given in the following: Lemma 4.27. Let m be a maximal element of L. Then the initial segment Mm = { x ∈ L | x ≤ m } is a maximal set in L with m as its unielement. Proof. Clearly Mm is a compatible set. Let y ∈ L be such that y ∼ x for all x ∈ Mm . In particular, y ∼ m implies that y ∧ m = m ∧ y and hence y ∨ m = m ∨ y = m since m is maximal. Thus y ≤ m. Hence y ∈ Mm . Thus Mm is a maximal set. Clearly m is the unielement of Mm . Theorem 4.28. An element m ∈ L is a maximal element if and only if m is the unielement of a maximal set in L.

References [1] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ. XXV, Providence, U.S.A. 1967. [2] G. Gratzer, General Lattice Theory, Academic Press, New York, Sanfransisco, 1978. [3] G.C. Rao, Almost Distributive Lattices, Doctoral Thesis (1980), Dept. of Mathematics, Andhra University, Visakhapatnam. [4] G.C. Rao, S. Ravi Kumar, Normal almost distributive lattices, Southeast Asian Bull. Math. 32 (2008) 831–841. [5] U.M. Swamy, G.C. Rao, Almost distributive lattices, Journa. Aust. Math. Soc (Series A) 31 77–91 (1981). [6] U.M. Swamy, G.C. Rao, G. Nanaji Rao, Pseudo-complementation on an almost distributive lattices, Southeast Asian Bull. Math. 24 95–104 (2000). [7] U.M. Swamy, , G.C. Rao, G. Nanaji Rao, Stone almost distributive lattices, Southeast Asian Bull. Math. 27 (2003) 513–526.