3_Rao.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 1, 2009, (58-72) ISSN 1307-5543 – www.ejpam.com Annulets in Almost Distributive Lattices G. C. Rao1∗ and M. Sambasiva Rao2 1 Department of Mathematics, Andhra University Visakhapatnam, Andhra Pradesh, India-530003 2 Department of Mathematics, M.V.G.R.College of Engineering Chintalavalasa, Vizianagaram, Andhra Pradesh, India-535003 Abstract. We introduce the concept of annulets in an Almost Distributive lattice(ADL) R with 0. We characterize both generalized stone ADL and normal ADL in terms of their annulets. We characterize ⋆-ADLs by means of their annulets. It is proved that the lattice A0(R) of all annulets of a generalized stone ADL R is a relatively complemented sublattice of the lattice I (R) of all ideals of R. Finally, it is proved thatA0(R) is relatively complemented iff R is sectionally ⋆-ADL. AMS subject classifications: 06D99, 06D15. Key words: Almost Distributive Lattice(ADL), Boolean algebra, dense elements, maximal element, Annihilator ideal, Annulet, normal ADL, ⋆-ADL, generalized stone ADL, Disjunctive ADL. 1. Introduction The concept of an Almost Distributive Lattice(ADL) was introduced by Swamy. U.M. and Rao.G.C [8] as a common abstraction to most of the existing ring theoretic and lattice the- oretic generalizations of a Boolean algebra. Later a more general class called ⋆-ADLs was introduced in the paper [10]. The characterization of ⋆-ADL by means of it’s dense elements was studied in [11]. In [5], Mandelker studied the properties of relative annihilators and ∗Corresponding author. Email addresses: g raomaths�yahoo. o.in (G. Rao),mssraomaths35�rediffmail. om (M. Rao) http://www.ejpam.com 58 c© 2009 EJPAM All rights reserved. G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 59 characterized the distributive lattice in terms of relative annihilators. In this paper the con- cept of Annulet as an ideal of the form (x]∗ = { a ∈ R | x ∧ a = 0 } in an ADL R with 0 is introduced, analogous to that in a distributive lattice[4]. It is proved that the set A0(R) of all annulets of an ADL R with 0 can be made into a distributive lattice and sublattice of the Boolean algebraA (R) of all annihilator ideals of R. We characterize the generalized stone ADL and normal ADL in terms of their annulets. We introduce a more general class of ADLs called disjunctive ADLs with suitable examples and prove that a disjunctive normal ADL is dually isomorphic to the latticeA0(R). We characterize ⋆-ADLs by means of their annulets. If R is a generalized stone ADL, then it is proved that the lattice A0(R) is a relatively complemented sublattice of the lattice I (R) of all ideals of R. Finally, it is proved thatA0(R) is relatively complemented iff R is sectionally ⋆-ADL. 2. Preliminaries An Almost Distributive Lattice (ADL) is an algebra (R,∨,∧) of type (2,2) satisfying 1. (x ∨ y)∧ z = (x ∧ z)∨ (y ∧ z) 2. x ∧ (y ∨ z) = (x ∧ y)∨ (x ∧ z) 3. (x ∨ y)∧ y = y 4. (x ∨ y)∧ x = x 5. x ∨ (x ∧ y) = x . for any x , y, z ∈ R. If R has an element 0 and satisfies 0∧ x = 0 and x∨0= x along with the above properties, then R is called an ADL with 0. Every non-empty set X can be regarded as an ADL as follows. Let x0 ∈ X . Define two G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 60 binary operations ∨,∧ on X by x ∨ y =    x if x 6= x0 y if x = x0 x ∧ y =    y if x 6= x0 x0 if x = x0 Then (X ,∨,∧, x0) is an ADL with x0 as zero element and is called a discrete ADL. If (R,∨,∧, 0) is an ADL, for any a, b ∈ R, define a ≤ b if and only if a = a ∧ b ( or equiv- alently, a ∨ b = b ), then ≤ is a partial ordering on R. Theorem 2.1. For any a, b, c ∈ R, 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 R 5. a ∧ b ∧ c = b ∧ a ∧ c 6. (a ∨ b)∧ c = (b ∨ a)∧ c 7. a ∧ b = 0⇔ b ∧ a = 0 8. a ∨ b = b ∨ a whenever a ∧ b = 0 9. a ∨ (b ∧ c) = (a ∨ b)∧ (a ∨ c) 10. a ∧ (a ∨ b) = a, (a ∧ b)∨ b = b, and a ∨ (b ∧ a) = a 11. a ≤ a ∨ b and a ∧ b ≤ b 12. a ∧ a = a and a ∨ a = a 13. 0∨ a = a and a ∧ 0= 0 14. If a ≤ c and b ≤ c then a ∧ b = b ∧ a and a ∨ b = b ∨ a 15. a ∨ b = a ∨ b ∨ a. An element m ∈ R is called maximal if it is maximal in the partial ordered set (R,≤). That is, for any x ∈ R, m ≤ x ⇒ m = x . Theorem 2.2. Let R be an ADL and m ∈ R. Then the following are equivalent: G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 61 1. m is a maximal element with respect to ≤ 2. m∨ x = m, for all x ∈ R 3. m∧ x = x , for all x ∈ R 4. x ∨m is maximal for all x ∈ R. A non-empty subset I of R is called an ideal(filter)of R if a ∨ b ∈ I(a ∧ b ∈ I) and a ∧ x ∈ I(x ∨ a ∈ I) whenever a, b ∈ I and x ∈ R. If I is an ideal of R and a, b ∈ R ,then a ∧ b ∈ I ⇔ b ∧ a ∈ I . The set I (R) of all ideals of R is a complete distributive lattice with least element {0} and the greatest element R under set inclusion in which, for any I , J ∈ I (R), I ∩ J is the infimum of I , J and the supremum is given by I ∨ J = { i ∨ j | i ∈ I , j ∈ J }. For any a ∈ R, (a] = { a ∧ x | x ∈ R } is the principal ideal generated by a. Similarly, for any a ∈ R, [a) = { x ∨ a | x ∈ R } is the filter generated by a. An ideal I of R is called a direct summand of R if there exists an ideal J in R such that I ∩ J = (0] and I ∨ J = R. Theorem 2.3. For any a, b ∈ R, we have the following: 1. (a]∨ (b] = (a ∨ b] = (b ∨ a] 2. (a]∩ (b] = (a ∧ b] = (b ∧ a] 3. [a)∨ [b) = [a ∧ b) = [b ∧ a) 4. [a)∩ [b) = [a ∨ b) = [b ∨ a) Thus the set P I (R) of all principal ideals of R is a sublattice of the distributive lattice I (R) of ideals of R. A proper ideal P of R is said to be prime if for any x , y ∈ R, x ∧ y ∈ P ⇒ either x ∈ P or y ∈ P. It is clear that a subset P of R is a prime ideal iff R− P is a prime filter. For any A⊆ R, A∗ = { x ∈ R | a ∧ x = 0 for all a ∈ A } is an ideal of R. We write (a]∗ for {a}∗. Then clearly (0]∗ = R and R∗ = (0]. An element a ∈ R is called dense if (a]∗ = (0]. The set of all dense elements of R is denoted by D. An ideal I of R is called dense if I∗ = (0]. An ADL R with 0 is called a ⋆-ADL [10], if for each x ∈ R, there exists an element x ′ ∈ R such that (x]∗∗ = (x ′]∗. R is a ⋆-ADL iff to each x ∈ R, there exists x ′ ∈ R such that x ∧ x ′ = 0 and x ∨ x ′ is dense. Every ⋆-ADL possesses a dense element. An ADL R with 0 is called relatively G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 62 complemented if each interval [a, b], a ≤ b, in R is a complemented lattice. An ideal I of R is called an annihilator ideal if I = I∗∗ , or equivalently, I = S∗ = { y ∈ R | y ∧ s = 0 for all s ∈ S } for some non-empty subset S of R. We denote the set of all annihilator ideals of R by A (R). The set A (R) forms a complete Boolean algebra with bounds {0},R and the complement of any I ∈A (R) is I∗ with respect to the operations ∧ and ∨ given by I ∧ J = I ∩ J and I ∨ J = (I∗ ∩ J∗)∗. 3. Annulets In this section, we introduce the concept of annulets in R and study some basic properties of these annulets. We prove charactarization theorems of a few algebraic structures with the help of their annulets. We begin with the following definition. Definition 3.1. Let R be an ADL with 0 and x ∈ R. Then define the annulet (x]∗ as follows: (x]∗ = { y ∈ R | x ∧ y = 0 } Clearly (x]∗ is an ideal in R and hence an annihilator ideal. Let us denoteA0(R) = { (x] ∗ | x ∈ R }. Annulets have many important properties. We give some of them in the following lemma which can be proved directly. Lemma 3.2. Let R be an ADL with 0 and x , y ∈ R. Then we have: 1. x ≤ y ⇒ (y]∗ ⊆ (x]∗ 2. (x ∧ y]∗ = (y ∧ x]∗ 3. (x ∨ y]∗ = (y ∨ x]∗ 4. (x ∨ y]∗ = (x]∗ ∩ (y]∗ 5. (x]∗ ∨ (y]∗ ⊆ (x ∧ y]∗. Note: Since each annulet is an annihilator ideal, we can have the following: (x]∗∨(y]∗ = � (x]∗∗ ∩ (y]∗∗ �∗ = � (x ∧ y]∗∗ �∗ = (x ∧ y]∗ G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 63 (x]∗ ∧ (y]∗ = (x]∗ ∩ (y]∗ = (x ∨ y]∗. Now we prove in the following theorem that the set A0(R) of all annulets of an ADL R forms a distributive lattice. Theorem 3.3. Let R be an ADL with 0. Then (A0(R),∩,∨) is a distributive lattice and a sublattice of the Boolean algebra A (R),∩,∨,∗ , (0],R � of annihilator ideals of R. A0(R) has the same greatest element R = (0]∗ as A (R) while A0(R) has the smallest element iff R possesses a dense element. Proof: Let (x]∗, (y]∗ ∈A0(R), where x , y ∈ R. Then 1. (x]∗ ∧ (y]∗ = (x]∗ ∩ (y]∗ = (x ∨ y]∗ ∈A0(R) and 2. (x]∗∨(y]∗ = (x ∧ y]∗ ∈A0(R). Hence A0(R) is a sublattice of A (R). Since A (R) is distributive, we have that A0(R) is also distributive. Clearly (0]∗ is the greatest element of A (R). Now for any (x]∗ ∈ A0(R), we get (x]∗∩(0]∗ = (x∨0]∗ = (x]∗ and (x]∗∨(0]∗ = (x∧0]∗ = (0]∗. It shows that (0]∗ is the greatest element inA0(R). Now, it remains to prove the final condition of the theorem. AssumeA0(R) has the smallest element, say (d]∗ where d ∈ R. Suppose x ∈ (d]∗. Then x ∧ d = 0. Since (d]∗ is the least element, we get (x]∗ = (x]∗∨(d]∗ = (x ∧ d]∗ = (0]∗ = R. Hence x = 0. Thus (d]∗ = (0]. Therefore d is a dense element in R. Conversely, suppose that R possesses a dense element, say d . So (d]∗ = (0]. Clearly (d]∗ ∈ A0(R). Now for any x ∈ R, consider (x]∗ ∩ (d]∗ = (x]∗ ∩ (0] = (0]. Also (x]∗∨(d]∗ = [(x]∗∗ ∩ (d]∗∗]∗ = [(x]∗∗ ∩ (0]∗]∗ = [(x]∗∗ ∩ R] ∗ = (x]∗∗∗ = (x]∗. Hence (d]∗ is the smallest element inA0(R). � The following definition of a normal ADL is taken from [7]. Definition 3.4. An ADL R with 0 is called normal ADL iff for all x , y ∈ R (x]∗ ∨ (y]∗ = (x ∧ y]∗. Swamy.U.M., Rao.G.C., Nanaji Rao.G.[9] and [10], have studied the properties of a psuedo- complemented ADL and later introduced the concept of stone ADL [10] as a psuedo-complemented G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 64 ADL R with 0, in which x∗∨x∗∗ = 0∗, for all x ∈ R. Now we give the definition of a generalized stone ADL in the following. Definition 3.5. An ADL R with 0 is called a generalized Stone ADL iff (x]∗ ∨ (x]∗∗ = R for each x ∈ R. Example 3.6. Let A = {0, a} and B = {0, b1, b2} be two discrete ADLs. Write R = A× B = {(0,0), (0, b1), (0, b2), (a, 0), (a, b1), (a, b2)}. Then (R,∨,∧, 0′) is an ADL where 0′ = (0,0), under point-wise operations. Now ((a, 0)]∗ ∨ ((a, 0)]∗∗ = {(0,0), (0, b1), (0, b2)} ∨ {(0,0), (a, 0)}= R. ((0, b1)] ∗ ∨ ((0, b1)] ∗∗ = {(0,0), (a, 0)} ∨ {(0,0), (0, b1), (0, b2)}= R. Also ((a, b1)] ∗ ∨ ((a, b1)] ∗∗ = {(0,0)} ∨ R= R. Hence (R,∨,∧, 0′) is a generalized stone ADL. We now characterize normal ADL and the generalized stone ADL in terms of annulets. Theorem 3.7. Let R be an ADL with 0. Consider the following conditions: (1). Each annulet is a direct summand of R (2). R is a generalized stone ADL (3). R is normal (4). A0(R) is a sublattice of the lattice I (R) of all ideals of R. Then (1) is equivalent to (2), (3) is equivalent to (4), and (2) implies (3). If R is a ⋆-ADL, then (4) implies (1). Proof: (1)⇒ (2): Let x ∈ R. Then by (1), there exists an ideal J of R such that (x]∗ ∩ J = (0] and (x]∗∨J = R. Now (x]∗∩J = (0] implies that J ⊆ (x]∗∗. Hence R= (x]∗∨J ⊆ (x]∗∨(x]∗∗. Thus R= (x]∗ ∨ (x]∗∗ ∀ x ∈ R. (2)⇒ (1): Assume that R is a generalized stone ADL. Let x ∈ R. We have always (x]∗ ∩ (x]∗∗ = (0]. By (2), we get (x]∗ ∨ (x]∗∗ = R. (2) ⇒ (3): Assume that R is a generalized stone ADL. Let x , y ∈ R. Always we have (x]∗ ∨ (y]∗ ⊆ (x ∧ y]∗. Let a ∈ (x ∧ y]∗. Then a ∧ x ∧ y = 0. ⇒ (a ∧ x ∧ y] = (0] G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 65 ⇒ (x]∩ (a ∧ y] = (0] ⇒ (a ∧ y] ⊆ (x]∗ ⇒ (x]∗∗ ⊆ (a ∧ y]∗ ⇒ (x]∗∗ ∩ (a ∧ y] = (0] ⇒ (x]∗∗ ∩ {(a]∩ (y]} = (0] ⇒ {(x]∗∗ ∩ (a]} ∩ (y] = (0] ⇒ (x]∗∗ ∩ (a]⊆ (y]∗ It is clear that (x]∗ ∩ (a]⊆ (x]∗ Thus we get that {(x]∗ ∩ (a]} ∨ {(x]∗∗ ∩ (a]} ⊆ (x]∗ ∨ (y]∗ ⇒ {(x]∗ ∨ (x]∗∗} ∩ (a]⊆ (x]∗ ∨ (y]∗ ⇒ R∩ (a]⊆ (x]∗ ∨ (y]∗ ( since R is a generalized stone ADL ) ⇒ (a]⊆ (x]∗ ∨ (y]∗ ⇒ a ∈ (x]∗ ∨ (y]∗ Hence (x ∧ y]∗ ⊆ (x]∗ ∨ (y]∗. Thus (x ∧ y]∗ = (x]∗ ∨ (y]∗. Therefore R is normal. Now we prove the equivalency of (3) and (4). (3)⇒ (4): Assume that R is normal. Let x , y ∈ R. We have always (x]∗ ∩ (y]∗ = (x ∨ y]∗ ∈ A0(R). Since R is normal, we get (x]∗ ∨ (y]∗ = (x ∧ y]∗ ∈ A0(R). Therefore A0(R) is a sublattice of I (R). (4)⇒ (3): Assume the condition (4). Let x , y ∈ R. Then by (4), (x]∗ ∨ (y]∗ = (z]∗, for some z ∈ R. Now (z]∗∗ = {(x]∗ ∨ (y]∗}∗ = (x]∗∗ ∩ (y]∗∗ = (x ∧ y]∗∗. Hence (x]∗ ∨ (y]∗ = (x ∧ y]∗. Therefore R is normal. (4) ⇒ (1): Suppose R is a ⋆− ADL. Assume that A0(R) is a sublattice of I (R). Let x ∈ R. Then there exists x ′ ∈ R such that (x]∗∗ = (x ′]∗. We have always (x]∗ ∩ (x]∗∗ = (0]. Now (x]∗ ∨ (x]∗∗ = (x]∗ ∨ (x ′]∗ = (z]∗, for some z ∈ R(by condition (4)). Hence (z]∗∗ = {(x]∗ ∨ (x ′]∗}∗ = (x]∗∗ ∩ (x ′]∗∗ = (x]∗∗ ∩ (x]∗∗∗ = (0]. Thus (x]∗ ∨ (x]∗∗ = (z]∗ = (0]∗ = R. Thus (x]∗ is a direct summand of R. � Definition 3.8. An ADL R with 0, is called disjunctive iff for all a, b ∈ R, (a]∗ = (b]∗ implies a = b. G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 66 Example 3.9. Let R= {0, a, b, c} be a set. Define ∨ and ∧ on R as follows: ∨ 0 a b c 0 0 a b c a a a a a b b a b a c c a a c ∧ 0 a b c 0 0 0 0 0 a 0 a b c b 0 b b 0 c 0 c 0 c Then clearly (R,∨,∧, 0) is an ADL with 0. Now, (a]∗ = (0], (b]∗ = {0, c} and (c]∗ = {0, b}. Thus x 6= y implies that (x]∗ 6= (y]∗ for all x , y ∈ R. Hence R is disjunctive. Theorem 3.10. A disjunctive ADL R is dually isomorphic toA0(R). Proof: Let R be a disjunctive ADL. Define a mapping Φ : R −→A0(R) by Φ(x) = (x]∗, for all x ∈ R. Clearly Φ is well-defined. (i). Let x , y ∈ R be such that Φ(x) = Φ(y). Then (x]∗ = (y]∗. Since R is disjunctive, we get that x = y. Therefore Φ is One-one. (ii). Let y ∈ A0(R). Then y = (x]∗, for some x ∈ R. Now for this x , Φ(x) = (x]∗ = y. Therefore Φ is onto. (iii). Let(x]∗, (y]∗ ∈A0(R), where x , y ∈ R. Then Φ(x ∧ y) = (x ∧ y]∗ = (x]∗∨(y]∗ = Φ(x)∨Φ(y). Again Φ(x ∨ y) = (x ∨ y]∗ = (x]∗ ∩ (y]∗ = Φ(x)∧Φ(y). Hence Φ is a dual isomorphism. � In an ADL R with 0, we know that a maximal element is always a dense element. Now we prove the converse in disjunctive ADL. Theorem 3.11. If R is a disjunctive ADL, then every dense element of R is a maximal element. Proof: Assume that R is disjunctive. Let m be a dense element of R. That is (m]∗ = (0]. For any x ∈ R, (m∨ x]∗ = (m]∗∩(x]∗ = (0]∩(x]∗ = (0] = (m]∗. Since R is disjunctive, we get that m∨ G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 67 x = m. Therefore m is a maximal element of R. � We now characterize a ⋆-ADl in terms of it’s lattice of annulets in the following theorem. Theorem 3.12. Let R be an ADL with 0. Then R is a ⋆-ADL iffA0(R) is a Boolean subalgebra ofA (R). Proof: Assume that R is a ⋆-ADL. Then R has a dense element, say d . Then (d]∗ = (0] is the least element and(0]∗ is the great- est element of the sublatticeA0(R) ofA (R). Let x ∈ R. Since R is a ⋆-ADL, there exists x ′ ∈ R such that (x]∗∗ = (x ′]∗. We now show that (x ′]∗ is the complement of (x]∗ inA0(R), for each x ∈ R. Now (x]∗∩(x ′]∗ = (x]∗∩(x]∗∗ = (0] and (x]∗∨(x ′]∗ = � (x]∗∗ ∩ (x ′]∗∗ �∗ = [(x]∗∗ ∩ (x]∗∗∗]∗ = [(x]∗∗ ∩ (x]∗]∗ = (0]∗. ThusA0(R) is a Boolean subalgebra ofA (R). Conversely assume that A0(R) is a Boolean subalgebra ofA (R). Let x ∈ R. Then (x]∗ ∈ A0(R). Since A0(R) is a subalgebra of A (R), there exists (y]∗ ∈ A0(R), with y ∈ R such that (x]∗ ∩ (y]∗ = (0] and (x]∗ ∨ (y]∗ = (0]∗. Now (x]∗ ∨ (y]∗ = (0]∗ ⇒ (x ∧ y]∗ = (0]∗ = R ⇒ x ∧ y = 0. Again, (x]∗ ∩ (y]∗ = (0] ⇒ (x ∨ y]∗ = (0] ⇒ x ∨ y is a dense element. Thus we proved that for each x ∈ R, there exists y ∈ R such that x ∧ y = 0 and x ∨ y is a dense element. Therefore R is a ⋆-ADL. � Definition 3.13. An ADL R with 0 is called sectionally ⋆-ADL iff for any x( 6= 0) ∈ R, the interval [0, x] is a ⋆-ADL. Before proving the next theorem, we need the following lemma. Lemma 3.14. Let I , J be two ideals in an ADL R. If I ∩ J and I ∨ J (i.e. The infimum and the supremum of I , J in the distributive lattice I (R)) are both principal ideals, then I , J are also principal ideals. Proof: Suppose I ∨ J = (a] and I ∩ J = (b], for some a, b ∈ R. Now a ∈ I ∨ J ⇒ a = c ∨ d for some c ∈ I and d ∈ J . Then c ∨ (b ∧ d) ∈ I . So that G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 68 ( c ∨ (b ∧ d) ]⊆ I . We now prove that I = ( c ∨ (b ∧ d) ]. Let x ∈ I . Then x ∈ I ∨ J = (a]. So x = a ∧ x = (c ∨ d)∧ x = (c ∧ x)∨ (d ∧ x) −→ (1). Now x ∈ I and d ∈ J ⇒ x ∧ d ∈ I ∩ J = (b]⇒ d ∧ x ∈ (b]. Hence d ∧ x = b ∧ d ∧ x −→ (2). From (1) and (2), we can obtain x = (c ∧ x) ∨ (b ∧ d ∧ x) = [c ∨ (b ∧ d)] ∧ x . Hence x ∈ ( c ∨ (b ∧ d) ]. Therefore I ⊆ ( c ∨ (b ∧ d) ]. By symmetry, we get that J is also a principal ideal. � Theorem 3.15. Let R be a generalized stone ADL. Then A0(R) is a relatively complemented sublattice of the lattice I (R) of all ideals of R. Proof: Let R be a generalized stone ADL. By theorem 3.7, A0(R) is a sublattice of I (R). So we can treate ∨ as ∨. SinceA0(R) is a distributive lattice with the greatest element (0]∗ = R, it is enough to prove that each interval of the form [I ,R], where I ∈ A0(R), is complemented. Let J = [(x]∗,R] be an interval inA0(R) and (y]∗ ∈ J . We have clearly (y]∗ ∩ (y]∗∗ = (0]. Since R is generalized stone ADL, we have (y]∗ ∨ (y]∗∗ = R for all y ∈ R. Now � (x]∩ (y]∗ ∨ � (x]∩ (y]∗∗ = (x]∩ � (y]∗ ∨ (y]∗∗ = (x]∩ R= (x]. Also � (x]∩ (y]∗ ∩ � (x]∩ (y]∗∗ = (x]∩ � (y]∗ ∩ (y]∗∗ = (x]∩ (0] = (0]. Thus we have that the infimum and the supremum of the ideals (x]∩ (y]∗ and (x]∩ (y]∗∗ are the principal ideals (0] and (x]. Therefore, by the above lemma, (x] ∩ (y]∗ and (x] ∩ (y]∗∗ must be the principal ideals. Suppose (x]∩ (y]∗ = (a] and (x]∩ (y]∗∗ = (b] for some a, b ∈ R. Now a ∈ (x]∩ (y]∗⇒ (a]⊆ (x]⇒ (x]∗ ⊆ (a]∗.Therefore (a]∗ ∈ J . Also (a] = (x] ∩ (y]∗ ⊆ (y]∗ ⇒ (y]∗∗ ⊆ (a]∗. Hence (y]∗ ∨ (y]∗∗ ⊆ (y]∗ ∨ (a]∗ ⇒ R ⊆ (a]∗ ∨ (y]∗. Thus R= (a]∗ ∨ (y]∗ −→ (1) Again (a]∗ ∩ (y]∗ ∩ (x] = (a]∗ ∩ (a] = (0]. Hence (a]∗ ∩ (y]∗ ⊆ (x]∗. But (x]∗ ⊆ (y]∗ and (x]∗ ⊆ (a]∗ imply that (x]∗ ⊆ (a]∗ ∩ (y]∗. Hence (a]∗ ∩ (y]∗ = (x]∗ −→ (2) From (1) and (2), (a]∗ is the required complement of (y]∗ in J . G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 69 HenceA0(R) is a relatively complemented sublattice of I (R). � Definition 3.16. Let I = [0, x] , 0 < x, be an interval in an ADL R with 0. For a ∈ I , define the annihilator (a]+ of a with respect to I as follows: (a]+ = { y ∈ I | y ∧ a = 0 }. Observe that (a]∗ ∩ I = (a]+. Lemma 3.17. For a ∈ I , the annihilator (a]+ is an ideal in I. Proof: Since 0 ∈ I and 0 ∧ a = 0, we get that 0 ∈ (a]+. Let r, s ∈ (a]+. Then r, s ∈ I and r ∧ a = s ∧ a = 0. Since r, s ∈ I , we get r ∨ s ∈ I , and (r ∨ s)∧ a = (r ∧ a)∨ (s ∧ a) = 0∨ 0= 0. Hence r ∨ s ∈ (a]+. Let y ∈ (a]+ and t ∈ I . Then y ∈ I and y ∧ a = 0. Hence y ∧ t ∈ I . Now (y ∧ t)∧ a = t ∧ y ∧ a = t ∧0= 0, which implies that y ∧ t ∈ (a]+. Thus (a]+ is an ideal of I . � Lemma 3.18. Let I = [0, x] , 0 < x, be an interval in an ADL R with 0. Then we have the following: (i). For a, b ∈ I , (a]+ ⊆ (b]+ implies (a]∗ ⊆ (b]∗. (ii). If z ∈ R, then (z]∗ ∩ I = (z ∧ x]+. Proof: (i). Let a, b ∈ I and suppose (a]+ ⊆ (b]+. Let t ∈ (a]∗. Then t ∧ a = 0 and t ∈ R ⇒ t ∧ x ∧ a = 0 and t ∧ x ∈ I , since x ∈ I . Which implies t ∧ x ∈ (a]+ ⊆ (b]+ ⇒ t ∧ x ∧ b = 0⇒ t ∧ b = 0, since t ∈ I = [0, x]. Hence t ∈ (b]∗. (ii). Let t ∈ (z]∗ ∩ I . Then t ∈ (z]∗ and t ∈ I . Hence t ∧ z = 0 and t ∈ I . Thus t ∧ z ∧ x = 0 and t ∈ I ⇒ t ∧ (z ∧ x) = 0 and t ∈ I ⇒ t ∈ (z ∧ x]+. Therefore (z]∗∩ I ⊆ (z∧ x]+. Again, let t ∈ (z∧ x]+, then t∧z∧ x = 0 and t ∈ I ⇒ z∧ t∧ x = 0 and t ∈ I ⇒ z∧t = 0 and t ∈ I ⇒ t ∈ (z]∗ and t ∈ I . Hence t ∈ (z]∗∩I . Thus (z∧x]+ ⊆ (z]∗∩I . Therefore (z]∗ ∩ I = (z ∧ x]+. � We now prove the characterization theorem of a sectionally ⋆-ADL in terms of it’s annulets. Before proving it, we can observe that if R is an ADL with 0 and I = [0, x], 0 < x for some x ∈ R, then A0(I) is a bounded distributive lattice ( with respect to the operations given in the theorem 3.3 ) with the greatest element I = (0]+ and the least element (x]+. G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 70 Theorem 3.19. Let R be an ADL with 0. ThenA0(R) is relatively complemented if and only if R is sectionally ⋆-ADL. Proof: Assume thatA0(R) is relatively complemented. We have to prove that each interval I = [0, x] in R is a ⋆−ADL. By theorem 3.12, it is enough to prove thatA0(I) is relatively complemented. Since A0(I) is a distributive lattice with the greatest element I = (0]+, it is enough to prove that each interval [J , I] , J ∈A0(I) is complemented. Choose a, b ∈ I such that (b]+ ∈ � (a]+, I � ⊆A0(I). Then (a]+ ⊆ (b]+ ⊆ I . By lemma 3.18(i), (a]∗ ⊆ (b]∗ ⊆ R. Since A0(R) is relatively complemented and (b]∗ ∈ [(a]∗,R], there exists an element c ∈ R such that (c]∗ ∈ [(a]∗,R] and (b]∗ ∩ (c]∗ = (a]∗ and (b]∗∨(c]∗ = R. Now (b]∗ ∩ (c]∗ = (a]∗ ⇒ (b]∗ ∩ (c]∗ ∩ I = (a]∗ ∩ I ⇒ [(b]∗ ∩ I] ∩ [(c]∗ ∩ I] = (a]∗ ∩ I ⇒ (b]+ ∩ (c]+ = (a]+ −→ (1) Secondly, (b]∗∨(c]∗ = R⇒ � (b]∗∨(c]∗ � ∩I = R∩I ⇒ [(b]∗ ∩ I]∨ [(c]∗ ∩ I] = I ⇒ (b]+∨(c]+ = I −→ (2) From (1) and (2), we get that (c]+ is the complement of (b]+ in � (a]+, I � . Hence � (a]+, I � is relatively complemented. Conversely assume that R is sectionally ⋆-ADL. Since A0(R) is a distributive lattice with the greatest element R, it is enough to prove that each interval [(a]∗,R] , (a]∗ ∈A0(R) is complemented. Let (b]∗ ∈ [(a]∗,R]. Therefore (a]∗ ⊆ (b]∗ ⊆ R. Consider the interval I = [0, b ∨ a]. Then by the hypothesis, I is a ⋆− ADL. So by theorem 3.12,A0(I) is complemented. Hence each interval � (a]+, I � , (a]+ ∈A0(I), where a ∈ I is complemented. We have by the lemma 3.18(ii), (a]∗ ∩ I = (a ∧ (b ∨ a)]+ and (b]∗ ∩ I = (b ∧ (b ∨ a)]+ = (b]+ ⊆ I , that is (b]+ ∈ � (a ∧ (b ∨ a)]+, I � . SinceA0(I) is complemented, there exists an element c ∈ I such that (b]+ ∩ (c]+ = (a ∧ (b ∨ a)]+ and (b]+∨(c]+ = I −→ (3) Now our claim is (b]∗ ∩ (c]∗ = (a]∗ and (b]∗∨(c]∗ = R. G. C. Rao and M. Sambasiva Rao / Eur. J. Pure Appl. Math, 2 (2009), (58-72) 71 Let x ∈ (b]∗ ∩ (c]∗. Then x ∈ (b]∗ and x ∈ (c]∗, implies b ∧ x = 0 and c ∧ x = 0 ⇒ x ∧ (b∨ a)∧ b = 0 and x ∧ (b ∨ a)∧ c = 0. ⇒ x ∧ (b∨ a) ∈ (b]+ and x ∧ (b ∨ a) ∈ (c]+, since x ∧ (b ∨ a) ∈ I . ⇒ x ∧ (b ∨ a) ∈ (b]+ ∩ (c]+ ⇒ x ∧ (b ∨ a) ∈ (a ∧ (b ∨ a)]+, by (3) ⇒ x ∧ (b ∨ a)∧ a ∧ (b ∨ a) = 0 ⇒ x ∧ a ∧ (b ∨ a)∧ (b ∨ a) = 0 ⇒ (x ∧ a)∧ (b ∨ a) = 0 ⇒ (b ∨ a)∧ (x ∧ a) = 0 ⇒ x ∧ (b ∨ a)∧ a = 0 ⇒ x ∧ a = 0 ⇒ x ∈ (a]∗ Hence (b]∗ ∩ (c]∗ ⊆ (a]∗ −→ (4) Conversely, let x ∈ (a]∗. Then x ∧ a = 0 ⇒ x ∧ a ∧ (b ∨ a)∧ (b ∨ a) = 0 ⇒ x ∧ (b ∨ a)∧ a ∧ (b ∨ a) = 0 ⇒ x ∧ (b ∨ a) ∈ (a ∧ (b ∨ a)]+, since x ∧ (b ∨ a) ∈ I . ⇒ x ∧ (b ∨ a) ∈ (b]+ ∩ (c]+, by (3) ⇒ x ∧ (b ∨ a) ∈ (b]+ and x ∧ (b ∨ a) ∈ (c]+ ⇒ x ∧ (b ∨ a)∧ b = 0 and x ∧ (b ∨ a)∧ c = 0. ⇒ x ∧ b = 0 and x ∧ c = 0, since c ∈ I = [0, b ∨ a]. ⇒ x ∈ (b]∗ and x ∈ (c]∗ ⇒ x ∈ (b]∗ ∩ (c]∗ Hence (a]∗ ⊆ (b]∗ ∩ (c]∗. −→ (5) From (4) and (5), we can obtain (b]∗ ∩ (c]∗ = (a]∗. Again from (3), we have (b]+∨(c]+ = I ⇒ (b ∧ c]+ = (b]+∨(c]+ = I ⇒ (b ∧ c]+ = I ⇒ b ∧ c = 0 REFERENCES 72 ⇒ (b ∧ c]∗ = (0]∗ = R ⇒ (b]∗∨(c]∗ = R Hence (c]∗ is the complement of (b]∗ in [(a]∗,R] . ThusA0(R) is relatively complemented. � ACKNOWLEDGEMENTS. The authors would like to thank the referee for his comments and valuable suggestions. References [1] Birkhoff. G. : Lattice Theory, Amer.Math.Soc.Colloq. XXV, Providence, (1967), U.S.A. [2] Burris. S., Sankappanavar, H.P. : A Cource in Univerasal Algebra, Springer Verlag, (1981). [3] Cornish.W.H. : Normal Lattices, J.Austral.Math.Soc., 14 (1972), 200-215. [4] Cornish.W.H.: Annulets and α- ideals in Distributive Lattices, J.Austral.Math.Soc., 15 (1973), 70-77. [5] Mandelker. M : Relative annihilators in lattices, Duke Math. J, 37 (1970), 377-386. [6] Rao.G.C. : Almost Distributive Lattices, Doctoral Thesis, Andhra University, Waltair, (1980). [7] Rao.G.C.and Ravikumar.S : Normal Almost Distributive Lattices, Southeast Asian Bulletin of math- ematics,(to appear). [8] Swamy.U.M., Rao.G.C. : Almost Distributive Lattices, J.Austral.Math.Soc. (Series A), 31 (1981), 77-91. [9] Swamy.U.M., Rao.G.C., Nanaji Rao.G. : Pseudo-Complementation on Almost Distributive Lattices, Southeast Asian Bulletin of mathematics, 24 (2000), 95-104. [10] Swamy.U.M., Rao.G.C., Nanaji Rao.G. : Stone Almost Distributive Lattices, Southeast Asian Bul- letin of mathematics, 27 (2003), 513-526. [11] Swamy.U.M., Rao.G.C., Nanaji Rao.G. : Dense Elements in Almost Distributive Lattices, Southeast Asian Bulletin of mathematics, 27 (2004), 1081-1088.