9_709_rao.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 4, 2010, 704-716 ISSN 1307-5543 – www.ejpam.com S−Linear Almost Distributive Lattices G.C. Rao1,∗, N. Rafi1, Ravi Kumar Bandaru2 1 Department of Mathematics, Andhra University, Visakhapatnam, Andhra Pradesh, India - 530003. 2 Department of Engineering Mathematics, GITAM University, Hyderabad Campus, Andhra Pradesh, India Abstract. The concept of an S−Linear ADL is defined and characterized in terms of the S−prime ideals and S−prime filters. Equivalent condition for an ADL R to become a (dually)B−relatively normal ADL in terms of minimal prime ideals(filters) and B−maximal ideals(filters) is obtained, where B is the Birkhoff centre of R. 2000 Mathematics Subject Classifications: 06D99 Key Words and Phrases: Almost Distributive Lattice (ADL), S−normal ADL, S−relatively normal ADL, S−linear ADL, uni subADL, Birkhoff centre, S−ideal, S−filter, Prime ideal, Prime filter. 1. Introduction The concepts of S−completely normal lattice and dually S−completely normal lattice were given by Cignoli [2]. The concept of an Almost Distributive Lattice (ADL) was introduced by Swamy and Rao [9] as a common abstraction of the existing ring theoretic and lattice theoretic generalizations of a Boolean algebra. The concept of an ideal in an ADL was introduced in [9] analogous to that in a distributive lattice and it was observed that the set PI(R) of all principal ideals of R forms a distributive lattice. This enables us to extend many existing concepts from the class of distributive lattices to the class of ADLs. In our paper [5], we introduced the concept of an S−normal ADL R, where S is a uni subADL of R and obtained necessary and sufficient conditions for an ADL R to become an S−normal ADL in terms of S−prime filters, S−maximal filters. B−normal ADLs were also studied, where B is the Birkhoff centre of R. In this paper, we define the concept of an S−relative annihilator of any two elements of R and characterize an S−normal ADL in terms of S−relative annihilators. ∗Corresponding author. Email addresses: g raomaths�yahoo. o.in (G. Rao), rafimaths�gmail. om (N. Rafi),ravimaths83�gmail. om (B. Kumar) http://www.ejpam.com 704 c© 2010 EJPAM All rights reserved. G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 705 We introduce the concepts of S−relatively normal ADL and dually S−relatively normal ADL. We characterize the (dually)S−relatively normal ADL in terms of S−prime filters(ideals). If B is the Birkhoff centre of R, then we define the concept of (dually)B−relatively normal ADL and characterize it in terms of minimal prime ideals(filters) and B−maximal ideals(filters)of R. 2. Preliminaries Definition 1 ([9]). An Almost Distributive Lattice with zero or simply ADL is an algebra (R,∨,∧, 0) of type (2,2,0) 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 6. 0∧ x = 0 7. x ∨ 0= x . Every non-empty set X can be regarded as an ADL as follows. Let x0 ∈ X . Define the 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 (where x0 is the zero) 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 equivalently, a ∨ b = b), then ≤ is a partial ordering on R. Theorem 1 ([9]). If (R,∨,∧, 0) is an ADL, 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. ∧ is associative in R 4. a ∧ b ∧ c = b ∧ a ∧ c 5. (a ∨ b)∧ c = (b ∨ a)∧ c 6. a ∧ b = 0⇔ b ∧ a = 0 G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 706 7. a ∨ (b ∧ c) = (a ∨ b)∧ (a ∨ c) 8. a ∧ (a ∨ b) = a, (a ∧ b)∨ b = b and a ∨ (b ∧ a) = a 9. a ≤ a ∨ b and a ∧ b ≤ b 10. a ∧ a = a and a ∨ a = a 11. 0∨ a = a and a ∧ 0= 0 12. If a ≤ c, b ≤ c then a ∧ b = b ∧ a and a ∨ b = b ∨ a 13. a ∨ b = (a ∨ b)∨ a. It can be observed that an ADL R satisfies almost all the properties of a distributive lattice except the right distributivity of ∨ over ∧, commutativity of ∨, commutativity of ∧. Any one of these properties make an ADL R a distributive lattice. That is Theorem 2 ([9]). Let (R,∨,∧, 0) be an ADL with 0. Then the following are equivalent: 1. (R,∨,∧, 0) is a distributive lattice 2. a ∨ b = b ∨ a, for all a, b ∈ R 3. a ∧ b = b ∧ a, for all a, b ∈ R 4. (a ∧ b)∨ c = (a ∨ c)∧ (b ∨ c), for all a, b, c ∈ R. As usual, an element m ∈ R is called maximal if it is a maximal element in the partially ordered set (R,≤). That is, for any a ∈ R, m≤ a⇒ m = a. Theorem 3 ([9]). Let R be an ADL and m ∈ R. Then the following are equivalent: 1. m is maximal with respect to ≤ 2. m∨ a = m, for all a ∈ R 3. m∧ a = a, for all a ∈ R 4. a ∨m is maximal, for all a ∈ R. As in distributive lattices [1, 3], a non-empty sub set I of an ADL R is called an ideal of R if a∨ b ∈ I and a∧ x ∈ I for any a, b ∈ I and x ∈ R. Also, a non-empty subset F of R is said to be a filter of R if a ∧ b ∈ F and x ∨ a ∈ F for a, b ∈ F and x ∈ R. The set I(R) of all ideals of R is a bounded distributive lattice with least element {0} and greatest element R under set inclusion in which, for any I , J ∈ I(R), I ∩ J is the infimum of I and J while the supremum is given by I ∨ J := {a ∨ b | a ∈ I , b ∈ J}. A proper ideal P of R is called a prime ideal if, for any x , y ∈ R, x ∧ y ∈ P ⇒ x ∈ P or y ∈ P. A proper ideal M of R is said to be maximal if it is not properly contained in any proper ideal of R. It can be G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 707 observed that every maximal ideal of R is a prime ideal. Every proper ideal of R is contained in a maximal ideal. For any subset S of R the smallest ideal containing S is given by (S] := {( n ∨ i=1 si)∧ x | si ∈ S, x ∈ R and n ∈ N}. If S = {s}, we write (s] instead of (S]. Similarly, for any S ⊆ R, [S) := {x ∨ ( n ∧ i=1 si) | si ∈ S, x ∈ R and n ∈ N}. If S = {s}, we write [s) instead of [S). Theorem 4 ([9]). For any x, y in R the following are equivalent: 1. (x]⊆ (y] 2. y ∧ x = x 3. y ∨ x = y 4. [y) ⊆ [x). For any x , y ∈ R, it can be verified that (x]∨ (y] = (x ∨ y] and (x]∧ (y] = (x ∧ y]. Hence the set PI(R) of all principal ideals of R is a sublattice of the distributive lattice I(R) of ideals of R. 3. S−relatively Normal ADLs If R is an ADL and S is a subADL with 0, then the concept of S−normality in R introduced in [5] and its properties were discussed. R. Cignoli [2] gave the concept of S−completely normal lattice. In this section we define the concept of S−relative normality in an ADL R through its principal ideal lattice PI(R). A subADL of an ADL with 0 carries the usual meaning where 0 is treated as a nullary operation. Through out this paper R represents an ADL and S stands for a subADL of R with 0. By a uni subADL of R we mean a subADL of R containing all maximal elements of R. In [8], the concept of relative annihilator in an ADL was given. If x , y ∈ R, then ⌊x , y⌋ = {a ∈ R | y ∧ a ∧ x = a ∧ x} is called a relative annihilator in R and ⌊x , 0⌋ = (x)∗ is the annihilator of x in R. Now we define the concept of an S−relative annihilator in R as follows. Definition 2. Let x , y ∈ R. Define ⌊x , y⌋S = {a ∈ S | y ∧ a ∧ x = a ∧ x}. We call ⌊x , y⌋S an S−relative annihilator. It can be observed that a ∈ ⌊x , y⌋S iff y = y ∨ (a ∧ x). Clearly ⌊x , y⌋S is an ideal of S. The following result can be verified easily. Lemma 1. Let x , y ∈ R. Then for any a ∈ S, a ∈ ⌊x , y⌋S iff x ∧ a ≤ y ∧ a. The following definition is taken from [5]. G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 708 Definition 3. Let S be a subADL of R. An ideal I of R is called an S-ideal of R if I is generated by the set I ∩ S(I = (I ∩ S]). An S−ideal I is called an S−prime ideal of R if I ∩ S is a prime ideal of S and S−maximal ideal if I ∩ S is a maximal ideal of S. It can be observed that every S−maximal ideal of R is an S−prime ideal. The concepts of S− filters, S−prime filters and S−maximal filters are defined analogously. Now, the following lemma can be verified easily. Lemma 2. Let R be an ADL, S a subADL of R and F1 a filter of S. Then the filter F of R is generated by F1 is S−filter of R and F1 = F ∩ S. We recall the following from [5]. Definition 4. Let R be an ADL with maximal elements and S a uni subADL of R. R is called S−normal if for any x , y ∈ R such that x ∧ y = 0 then there exist elements a, b ∈ S such that x ∧ a = 0= y ∧ b and a ∨ b is a maximal element. In the following theorem, we characterize the S−normal ADL in terms of S−relative an- nihilators. Theorem 5. Let R be an ADL with maximal elements and S a uni subADL of R. Then the following conditions are equivalent: 1. R is S−normal 2. ⌊x , y⌋S ∨ ⌊y, x⌋S = S, for any x , y ∈ R with x ∧ y = 0 3. For any prime filter F of S and for any x , y ∈ R with x ∧ y = 0, there exists a ∈ F such that x ∧ a and y ∧ a are comparable. Proof. (1)⇒ (2) : Assume that R is an S−normal ADL. Let x , y ∈ R such that x ∧ y = 0. Then there exist a, b ∈ S such that a ∧ x = 0 = b ∧ y and a ∨ b is a maximal element. That implies y ∧ a ∧ x = a ∧ x = 0= x ∧ b ∧ y = b ∧ y. Therefore ⌊x , y⌋S ∨ ⌊y, x⌋S = S. (2) ⇒ (3) : Let F be any prime filter of S and x , y ∈ R such that x ∧ y = 0. Then ⌊x , y⌋S ∨ ⌊y, x⌋S = S. Let m be any maximal element in S. Then m = a ∨ b, for some a ∈ ⌊x , y⌋S and b ∈ ⌊y, x⌋S. That implies a∧ x = y∧a∧ x = 0 and b∧ y = x∧ b∧ y = 0. Since a ∨ b ∈ F, we get either a ∈ F or b ∈ F. Suppose a ∈ F. Since a ∈ ⌊x , y⌋S , we get x ∧ a ≤ y ∧ a. Thus there is an element a ∈ F such that x ∧ a and y ∧ a are comparable. Similarly, we get x ∧ b and y ∧ b are comparable, if b ∈ F. (3)⇒ (1) : Let x , y ∈ R such that x ∧ y = 0. Suppose that ((x)∗ ∩ S) ∨ ((y)∗ ∩ S) 6= S. Then there exists a maximal ideal M of S such that ((x)∗ ∩ S) ∨ ((y)∗ ∩ S) ⊆ M . That implies S \ M is a prime filter of S. By (3), there exists x ∈ S \ M such that x ∧ a and y ∧ a are comparable. Suppose x ∧ a ≤ y ∧ a. Then x ∧ a = x ∧ a ∧ y ∧ a = 0. Then a ∈ ⌊x , y⌋S ∩ (S \ M), which is a contradiction. Therefore ((x)∗ ∩ S) ∨ ((y)∗ ∩ S) = S. Hence R is S−normal. G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 709 In [7], the concept of relatively normal ADL was given as follows. Definition 5. Let R be an ADL with maximal elements. Then R is called relatively normal if for any x , y ∈ R, there exist a, b ∈ R such that y ∧ a ∧ x = a ∧ x , x ∧ b ∧ y = b ∧ y and a ∨ b is a maximal element. The following definition is taken from Cignoli [2]. Definition 6. Let (L,∨,∧, 0,1) be a bounded distributive lattice and S a sublattice of L contain- ing 0 and 1. Then L is called S−completely normal, if for any x , y ∈ L, there exist a, b ∈ S such that x ∧ a ≤ y, y ∧ b ≤ x and a ∨ b = 1. Now we define the concept of an S−relatively normal ADL in the following. Definition 7. Let R be an ADL with maximal elements and S a uni subADL of R. R is called S−relatively normal if P I(R) is P I(S)−completely normal lattice. The following lemma can be verified directly. Lemma 3. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is S−relatively normal if and if only for any x , y ∈ R, there exist a, b ∈ S such that y ∧ a ∧ x = a ∧ x , x ∧ b ∧ y = b ∧ y and a ∨ b is a maximal element. Example 1. Let A be a discrete ADL and B a Boolean algebra. Then R = A× B is an ADL. Let D be a subADL of A containing at least two elements. Then S = D×B is a subADL of R. Let x , y ∈ R. Then x = (x1, x2) and y = (y1, y2). Let t be any non-zero element of D. Suppose x1, y1 6= 0. Write a = (t, y2∨x ′2) and b = (t, x2∨y ′2). Now, y∧a∧x = (y1, y2)∧(t, y2∨x ′2)∧(x1, x2) = (y1∧ t∧x1, y2∧(y2∨x ′2)∧x2) = (x1, y2∧x2) = a∧x and x∧b∧ y = (x1∧ t∧ y1, x2∧(x2∨ y ′2)∧ y2) = (y1, x2∧ y2) = b∧ y. Also a∨ b = (t, 1). Now, suppose x1 = 0 and y1 6= 0. Take a = (t, y2 ∨ x ′2) and b = (0, x2 ∨ y ′2). Now, y ∧ a ∧ x = (y1 ∧ t ∧ 0, y2 ∧ (y2 ∨ x ′2)∧ x2) = (0, y2 ∧ x2) = a ∧ x and x ∧ b ∧ y = (0∧ 0∧ y1, x2 ∧ (x2 ∨ y ′2)∧ y2) = (0, x2 ∧ y2) = b ∧ y. Clearly a ∨ b = (t, 1). Thus R is an S−relatively normal ADL. Lemma 4. Let R be an ADL with maximal elements and S a uni subADL of R. If R is S−relatively normal, then, for each pair a, b ∈ S such that a < b, the segment [a, b] is an S ∩ [a, b]−normal lattice. Proof. Let x , y ∈ [a, b] such that x ∧ y = a. Since R is S−relatively normal, there exist c, d ∈ S such that y ∧ c ∧ x = c ∧ x , x ∧ d ∧ y = d ∧ y and c ∨ d is a maximal element. Now, take c1 = a ∨ (c ∧ b) and d1 = a ∨ (d ∧ b). Clearly c1, d1 ∈ [a, b] ∩ S. Now, c1 ∧ x = (a ∨ (c ∧ b))∧ x = (a ∧ x)∨ (c ∧ b ∧ x) = a ∨ (c ∧ x) = a ∨ (c ∧ y ∧ x) = a ∨ (c ∧ a) = a and d1∧ y = (a∨(d∧ b))∧ y = (a∧ y)∨(d∧ b∧ y) = a∨(d∧ y) = a∨(d∧ x∧ y) = a∨(d∧a) = a. Clearly c1 ∨ d1 = b. Therefore [a, b] is S ∩ [a, b]−normal lattice. The following two results can be verified easily. Lemma 5. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is S−relatively normal if and only if for any x , y ∈ R, ⌊x , y⌋S ∨ ⌊y, x⌋S = S. G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 710 Lemma 6. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is S−relatively normal if and only if for any prime filter F of S and for any x , y ∈ R, there ex- ists a ∈ F such that x ∧ a and y ∧ a are comparable. Theorem 6. Let R be an ADL with maximal elements, S a uni subADL of R, F an S−filter of R and K a non-empty subset of R, which is closed under the operation join such that F ∩ K = ;. Then there exists an S−prime filter P of R such that F ⊆ P and P ∩ K = ;. Theorem 7. Let R be an ADL with maximal elements and S a uni subADL of R. Then the following conditions are equivalent: 1. R is S−relatively normal 2. For each pair x , y ∈ R, there is no proper ideal of S contain both ⌊x , y⌋S and ⌊y, x⌋S 3. The set of all filters of R that contain a given S−prime filter of R form a chain 4. The set of all prime filters of R that contain a given S−prime filter of R form a chain 5. Any proper filter of R that contain a given S−prime filter of R is prime. Proof. (1)⇒ (2) : It follows from lemma 5. (2)⇒ (3) : Assume (2). Suppose P is an S−prime filter of R and F1, F2 are two filters of R such that P ⊆ F1 and P ⊆ F2. Suppose F1 * F2 and F2 * F1. Choose x ∈ F1 \ F2 and y ∈ F2 \ F1. Let a ∈ ⌊x , y⌋S . Then y ∧ a ∧ x = a ∧ x . Suppose a /∈ S \ (P ∩ S). Then a ∈ P ∩ S. That implies a ∈ F1 and x ∈ F1. Hence a∧ x ∈ F1. Thus y ∨ (a∧ x) = y ∈ F1, which is a contradiction. Therefore a ∈ S \ (P ∩ S). Hence ⌊x , y⌋S ⊆ S \ (P ∩ S) and similarly, we have ⌊y, x⌋S ⊆ S \ (P ∩ S). Since S \ (P ∩ S) is a prime ideal of S, this is a contradiction. (3)⇒ (4) : Clear. (4)⇒ (5) : Assume (4). Let P be an S−prime filter of R and F a proper filter of R such that P ⊆ F. Suppose F is not prime filter of R. Then there exist a, b ∈ R such that a /∈ F, b /∈ F and a ∨ b ∈ F. Then there exist prime filters Pa, Pb of R such that a /∈ Pa, b /∈ Pb and F ⊆ Pa ∩ Pb. Since a∨ b ∈ Pa ∩ Pb, we get b ∈ Pa and a ∈ Pb. Therefore Pa * Pb and Pb * Pa, which is a contradiction. Hence F is a prime filter of R. (5) ⇒ (1) : Assume (5). Let x , y ∈ R. Suppose ⌊x , y⌋S ∨ ⌊y, x⌋S 6= S. Let m be any maximal element in R. Then m /∈ ⌊x , y⌋S ∨⌊y, x⌋S and hence there exists an prime filter P ′ of S such that (⌊x , y⌋S∨⌊y, x⌋S)∩P ′ = ;. So that ⌊x , y⌋S∩P ′ = ; and ⌊y, x⌋S∩P ′ = ;. Let P be the filter of R generated by P ′. By the lemma 2, we get that P is an S−prime filter of R and P ′ = P ∩ S. If 0 ∈ P ∨ [x ∨ y), then 0 = p ∧ (x ∨ y) and hence p ∧ x = 0 and p ∧ y = 0. Since p ∈ P, there exists s ∈ P ∩ S = P ′ such that p ∨ s = p. Now, we prove that the filter P ∨ [x ∨ y) is a proper filter of R. Now, s ∧ x = p ∧ s ∧ x = 0. So G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 711 that s ∈ ⌊x , y⌋S ∩ P ′, which is a contradiction. Therefore P ∨ [x ∨ y) is a proper filter of R containing P. By our assumption, P ∨ [x ∨ y) is a prime filter of R. Without loss of generality, suppose x ∈ P ∨ [x ∨ y). Then x = t ∧ (x ∨ y), for some t ∈ P. Since t ∈ P, there exists s1 ∈ P∩S such that t∨s1 = t. Now, s1∧ x = s1∧ t∧(x∨ y) = (s1∧ x)∨(s1∧ y) and hence s1 ∧ y = s1 ∧ x ∧ s1 ∧ y = x ∧ s1 ∧ y. That implies s1 ∈ ⌊y, x⌋S ∩ P, which is a contradiction. Therefore ⌊x , y⌋S ∨ ⌊y, x⌋S = S. Corollary 1. Let R be an ADL with maximal elements and S1,S2 uni subADLs of R such that S1 ⊆ S2. Then the following conditions are equivalent: 1. R is S1−relatively normal 2. R is S2−relatively normal and the filters generated in S2 by prime filters of S1 are prime. Proof. (1) ⇒ (2) : Assume that R is S1−relatively normal. Clearly R is S2−relatively normal and S2 is S1−relatively normal. Let P be a prime filter of S1. We have to prove that [P) is an S1−prime filter of S2, where [P) = {s ∨ a | s ∈ S2 and a ∈ P}. Let x , y ∈ [P). Then x = s1 ∨ a1 and y = s2 ∨ a2, for some s1, s2 ∈ S2 and a1, a2 ∈ P. Now, x ∧ y = (s1∨a1)∧(s2∨a2) = (s1∧(s2∨a2))∨(a1∧(s2∨a2)) = (s1∧(s2∨a2))∨((a1∧s2)∨(a1∧a2)) and hence (x∧ y)∧(a1∧a2) = ((s1∧(s2∨a2))∨((a1∧s2)∨(a1∧a2)))∧(a1∧a2) = (a1∧a2). Thus x ∧ y = (x ∧ y) ∨ (a1 ∧ a2) and hence x ∧ y ∈ [P). Let x ∈ [P) and r ∈ S2. Then x = s ∨ a, for some s ∈ S2 and a ∈ P. Now, (r ∨ x)∧ a = (r ∨ (s ∨ a))∧ a = a and hence r ∨ x = (r ∨ x)∨ a. Therefore r ∨ x ∈ [P). Hence [P) is a filter of S2. Let x ∈ [P). Then x = s ∨ a, for some s ∈ S2 and a ∈ P. Now, x ∨ a = (s ∨ a)∨ a = s ∨ a = x . Hence [P) is an S1−filter of R. Let a, b ∈ S1 such that a ∨ b ∈ [P)∩ S1. Then a ∨ b = s ∨ x , for some s ∈ S2 and x ∈ P. Now, x = (a ∨ b) ∧ x = (a ∧ x) ∨ (b ∧ x) ∈ P (since x ∈ P). That implies either a ∧ x ∈ P or b ∧ x ∈ P. Suppose a ∧ x ∈ P. Then a ∧ x ∈ [P). That implies a ∨ (a ∧ x) ∈ [P)∩ S1. Hence a ∈ [P)∩ S1. Thus [P) is an S1− prime filter of S2. Since S2 is S1−relatively normal, [P) is a prime filter of S2. (2) ⇒ (1) : Assume that R is S2−relatively normal and the filters generated in S2 by prime filters of S1 are prime. Let P be an S1−prime filter of R. Let F be a proper filter of R such that P ⊆ F. Clearly P is an S2−filter of R. We have to prove that [P∩S1) = P∩S2. Let a ∈ [P ∩S1). Then a = s∨ x , for some s ∈ S2 and x ∈ P ∩S1. That implies a ∈ P ∩S2. Therefore [P ∩ S1) ⊆ P ∩ S2. Let a ∈ P ∩ S2. Then there exists s ∈ P ∩ S1 such that a ∨ s = a. That implies a ∈ [P ∩ S1). Hence P ∩ S2 is a prime filter of S2. That implies P is an S2−prime filter of R. Therefore F is prime filter of R. Thus R is an S1−relatively normal ADL. Corollary 2. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is S−relatively normal if and only if R is relatively normal and the S−prime filters of R are prime. G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 712 Proof. Take S1 = S and S2 = R in the above corollary. Let R be an ADL and F a filter in R. Then the relation ψ(F) = {(x , y) ∈ R×R | x∧ t = y∧ t, for some t ∈ F} is a congruence relation on R and the set R/ψ(F) = {x/ψ(F) | x ∈ R} is an ADL. Let ∏ be the natural homomorphism from R onto R/ψ(F) defined by ∏ (x) = x/ψ(F) for all x ∈ R. Theorem 8. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is S−relatively normal if and only if R/ψ(F) is a chain, for each prime filter F of S. Proof. Assume that R is S−relatively normal. Let x/ψ(F), y/ψ(F) ∈ R/ψ(F). Since x , y ∈ R, by theorem 6, there exists a ∈ F such that x ∧ a and y ∧ a are comparable. With out loss of generality, suppose x∧a ≤ y∧a. Then x∧a = x∧a∧ y∧a = x∧ y∧a. That implies (x , x∧ y) ∈ ψ(F) and hence x/ψ(F) = (x ∧ y)/ψ(F) = x/ψ(F)∧ y/ψ(F). Therefore x/ψ(F) ≤ y/ψ(F). Hence R/ψ(F) is a chain. Conversely, assume that R/ψ(F) is a chain. Let x , y ∈ R. Then x/ψ(F), y/ψ(F) ∈ R/ψ(F). Since R/ψ(F) is a chain, x/ψ(F), y/ψ(F) are comparable. With out loss of generality, suppose x/ψ(F) ≤ y/ψ(F). Then x/ψ(F) = x/ψ(F) ∧ y/ψ(F). That implies (x , x ∧ y) ∈ ψ(F). Then x ∧ a = x ∧ y ∧ a, for some a ∈ F. Therefore x ∧ a ≤ y ∧ a. Thus R is an S−relatively normal. The following result follows directly from the above theorem. Theorem 9. Each S−relatively normal ADL is a subdirect product of the bounded chains R \ P, where P runs through the set of all prime ideals of S. 4. Dually S−relatively Normal ADLs The concept of a dually S−completely normal lattices was given by Cignoli [2]. In this section we define the concept of dually S−relative normality in an ADL R through its principal filter lattice PF(R). We begin with the following. Definition 8. Let R be an ADL, S a uni subADL of R and x , y ∈ R. We define ⌈x , y⌉S = {a ∈ S | (x ∨ a)∨ y = x ∨ a}. We call ⌈x , y⌉S an S−relative dual annihilator. It can be observed that a ∈ ⌈x , y⌉S iff y = (x ∨ a)∧ y. Clearly ⌈x , y⌉S is a filter of S. The usual lattice theoretic duality principle doesn’t hold in ADLs. For example, in an ADL R, ∧ is right distributive over ∨ but ∨ is not right distributive over ∧. However, we get that the dual of many results of section 3, hold good in dually S− relatively normal ADLs. For this reason we give only statements of these results. Lemma 7. Let R be an ADL with maximal elements and S a uni subADL of R. If m1, m2 are two maximal elements in R, then for any x ∈ R, ⌈x , m1⌉S = ⌈x , m2⌉S. Lemma 8. Let P be any prime ideal of S. For any x , y ∈ R, if y ∈ P ∨ (x], then P ∩ ⌈x , y⌉S is non-empty. G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 713 Definition 9. Let R be an ADL with maximal elements and S a uni subADL of R. R is called dually S−normal if for any x , y ∈ R with x ∨ y is a maximal element in R, then there exist a, b ∈ R such that x ∨ a, y ∨ b are maximal elements and a ∧ b = 0. Theorem 10. Let R be an ADL with maximal elements and S a uni subADL of R. Then the following are equivalent: 1. R is dually S−normal 2. ⌈x , y⌉S ∨ ⌈y, x⌉S = S, for any x , y ∈ R with x ∨ y is a maximal element. The following definition is taken from [8]. Definition 10. Let R be an ADL with maximal elements. Then R is called dually relatively normal if for any x , y ∈ R there exist a, b ∈ R such that (x ∨ a) ∨ y = x ∨ a, (y ∨ b) ∨ x = y ∨ b and a ∧ b = 0. The following definition is taken from Cignoli [2]. Definition 11. Let (L,∨,∧, 0,1) be a bounded distributive lattice and S a sublattice of L con- taining 0 and 1. Then L is called dually S−completely normal, if for any x , y ∈ L, there exist a, b ∈ S such that x ∨ a ≥ y, y ∨ b ≥ x and a ∧ b = 0. Now we define the concept of dually S−relatively normal ADL in the following. Definition 12. Let R be an ADL with maximal elements and S a uni subADL of R. R is called dually S−relatively normal if PF(R) is dually PF(S)−completely normal lattice. Lemma 9. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is dually S−relatively normal if and only if for any x , y ∈ R, there exist a, b ∈ R such that (x ∨ a)∨ y = x ∨ a, (y ∨ b)∨ x = y ∨ b and a ∧ b = 0. Lemma 10. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is dually S−relatively normal if and only if for any x , y ∈ R, ⌈x , y⌉S ∨ ⌈y, x⌉S = S. Theorem 11. Let R be an ADL with maximal elements and S a uni subADL of R. Then the following conditions are equivalent: 1. R is dually S−relatively normal 2. For each pair x , y ∈ R, there is no proper filter of S containing both ⌈x , y⌉S and ⌈y, x⌉S 3. The set of all ideals of R that contain a given S−prime ideal of R form a chain 4. The set of all prime ideals of R that contain a given S−prime ideal of R form a chain 5. Any proper ideal of R that contain a given S−prime ideal of R is a prime. Corollary 3. Let R be an ADL with maximal elements and S1,S2 uni subADLs of R such that S1 ⊆ S2. Then the following conditions are equivalent: G. Rao, N. Rafi and B. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 704-716 714 1. R is dually S1−relatively normal 2. R is dually S2−relatively normal and the ideals generated in S2 by prime ideals of S1 are prime. Corollary 4. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is dually S−relatively normal if and only if R is dually relatively normal and the S−prime ideals of R are prime. Proof. Take S1 = S and S2 = R in the above corollary. Definition 13. Let R be an ADL with maximal elements. R is called relatively normal if for any x , y ∈ R, there exist a, b ∈ R such that y ∧ a ∧ x = a ∧ x , x ∧ b ∧ y = b ∧ y and a ∨ b is a maximal element. R is called dually relatively normal if for any x , y ∈ R, there exist a, b ∈ R such that (x ∨ a)∨ y = x ∨ a, (y ∨ b)∨ x = y ∨ b and a ∧ b = 0. Definition 14. Let R be an ADL with maximal elements. Then R is called a linear ADL if R is both relatively normal and dually relatively normal. If S is a uni subADL of R, then R is called an S−linear ADL if R is both S−relatively normal and dually S−relatively normal. The following theorem can be verified easily. Theorem 12. Let R be an ADL with maximal elements and S a uni subADL of R. Then R is S−linear if and only if 1. R is a linear ADL 2. The S−prime filters of R are prime in R 3. The S−prime ideals of R are prime in R. Definition 15. Let R be an ADL with maximal elements. Then B = {a ∈ R | there exists b ∈ R such that a ∧ b = 0 and a ∨ b is maximal} is called the Birkhoff centre of R and (B, ∨, ∧) is a uni sub ADL of R which is also a relatively complemented ADL [10]. If a ∈ B, then an element b ∈ R with the property a ∧ b = 0 and a ∨ b is maximal is called a complement of a in B. It was observed in [7] that every relatively complemented ADL is a normal ADL and hence B is normal. We conclude this paper with the following characterization theorem. Theorem 13. Let R be an ADL with maximal elements and B the Birkhoff centre of R. Then the following conditions are equivalent: 1. R is B−relatively normal REFERENCES 715 1.′ R is dually B−relatively normal 2. Given x , y ∈ R, there is a ∈ B and a complement a′ of a such that y ∧ a ∧ x = a ∧ x and x ∧ a′ ∧ y = a′ ∧ y 2.′ Given x , y ∈ R, there is a ∈ B a complement a′ of a such that x ∨ a ∨ y = x ∨ a and y ∨ a′ ∨ x = y ∨ a′ 3. R is a linear ADL and the minimal prime ideals of R are B−maximal ideals of R 3.′ R is linear ADL and the minimal prime filters of R are B−maximal filters of R. Proof. (1)⇒ (2) : Assume (1). Let x , y ∈ R. Then there exist a, b ∈ B such that y ∧ a ∧ x = a∧x , x∧b∧y = b∧y and a∨b is a maximal element. Since a ∈ B, there exists c ∈ R such that a∧ c = 0 and a∨ c is a maximal element. Now, a∨ (c∧ b) = (a∨ c)∧ (a∨ b) = a∨ b and a ∧ c ∧ b = 0. So that c ∧ b(= a′ say ) is a complement of a in B and x ∧ a′ ∧ y = x ∧ c ∧ b ∧ y = c ∧ x ∧ b ∧ y = c ∧ b ∧ y = a′ ∧ y. (2)⇒ (2′) : Assume (2). Let x , y ∈ R. Then by our assumption there exists a ∈ B and a complement a′ of a in B such that y ∧ a ∧ x = a ∧ x and x ∧ a′ ∧ y = a′ ∧ y. Now, (x ∨ a)∧ y = ((x ∨ (a′ ∧ y)∨ a)∧ y = (x ∨ a ∨ a′)∧ (x ∨ a ∨ y)∧ y = (a ∨ a′)∧ y = y. Therefore (x ∨ a)∨ y = x ∨ a. Similarly, (y ∨ a′)∨ x = y ∨ a′. (2′) ⇒ (2) : Assume (2′). Let x , y ∈ R. Then there exists a ∈ B and a complement a′ of a in B such that (x ∨ a) ∨ y = x ∨ a and (y ∨ a′) ∨ x = y ∨ a′. Now, y ∨ (a ∧ x) = y ∨ (a ∧ (y ∨ a′)∧ x) = y ∨ ((a∧ y ∧ x)∨ (a∧ a′ ∧ x)) = y ∨ (a ∧ y ∧ x) = y. Therefore y ∧ (a ∧ x) = a ∧ x . Similarly, x ∧ (a′ ∧ y) = a′ ∧ y. (2)⇒ (3) : Assume (2). Since (2) and (2′) are equivalent, R is a linear ADL. Let P be a minimal prime ideal of R and x ∈ P. Then there exists y ∈ R \ P such that x ∧ y = 0. By (2), there exists a ∈ B and a complement a′ of a such that a ∧ x = y ∧ (a ∧ x) = 0 and a′ ∧ y = x ∧ (a′ ∧ y) = 0. So that a′ ∧ y ∈ P and hence a′ ∈ P. Now, x = (a ∨ a′)∧ x = (a ∧ x) ∨ (a′ ∧ x) = a′ ∧ x . Therefore P is B−ideal of R and hence P is a B−maximal ideal of R, since B is a relatively complemented ADL. Similarly, we get that (2′)⇒(3′). References [1] Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloq. Publ. XXV, Providence. 1967. [2] Cignoli, R.: The lattice of global sections of sheaves of chains over Boolean spaces. Algebra Univesalis, 8, 357-373. 1978. [3] Gratzer, G.: General Lattice Theory. Academic Press, New York, Sanfransisco. 1978. REFERENCES 716 [4] Rao, G.C.: Almost Distributive Lattices. Doctoral Thesis, Dept. of Mathematics, Andhra University, Visakhapatnam. 1980. [5] Rao, G.C. Rafi, N. and Ravi Kumar Bandaru.: S−ideals in Almost Distributive Lattices. Accepted for publication in Southeast Asian Bulletin of Mathematics. [6] Rao, G.C. and Ravi Kumar, S.: Minimal prime ideals in an ADL. Int. J. Contemp. Sciences, 4, 475-484. 2009. [7] Rao, G.C. and Ravi Kumar, S.: Normal Almost Distributive Lattices. Southeast Asian Bullettin of Mathematics, 32, 831-841. 2008. [8] Ravi Kumar, S.: Normal Almost Distributive Lattices. Doctoral Thesis, Dept. of Mathe- matics, Andhra University, Visakhapatnam. 2009. [9] Swamy, U.M. and Rao, G.C.: Almost Distributive Lattices. J. Aust. Math. Soc. (Series A), 31, 77-91. 1981. [10] Swamy, U.M. and Ramesh, S.: Birkhoff Centre of ADL. Int. J. Algebra, 3, 539-546. 2009.