CHAPTER - 6 DISTRIBUTIVE CONVEX SUBLATTICE

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1 CHAPTER - 6 DISTRIBUTIVE CONVEX SUBLATTICE 0 Introduction: The concept of standard convex sublattice or standard sublattice (i.e., generalization of standard ideals for convex sublattice) has been introduced and studied by Fried, E and Schmidt, E.T [1]. In this chapter we give a generalization of distributive ideals for Convex sublattice called distributive Convex sublattice or Distributive sublattice.it is proved that many important properties of distributive ideals are also valid for distributive sublattice.we also establish the connections of distributive Convex sublattices with Congruence classes of lattices. We prove that an ideal D of lattice L is distributive if and only if it is a distributive sublattice of a lattice L. Also we prove an ideal D is distributive sublattice of a lattice if and only if a binary relation θ D is a Congruence relation. We shall denote and as set theoretical operations as addition and juxtaposition. The lattice theoretical operations explicitly to represent X Y, we write X + Y and to represent X Y, we write X.Y, where X, Y in a lattice. 94

2 1 Distributive Convex sublattices : 1.1 Definition: A non empty subset S of a lattice L is said to be a sublattice of L if for any a, b in S, a b and a b are also in S. 1.2 Note: Every sublattice of a lattice is a lattice with respect to operations defined in the original lattice. 1.3 Definition: A sublattice S of a lattice L, is called a convex sublattice, if for all a,b in S, [a b, a b] S. 1.4 Note: A sublattice S of a lattice L is a convex sublattice if and only if for all a,b in S, a b, [a,b] S. 1.5 Note: A convex sublattice generated by a subset A of the lattice L, will be denoted by < A>. 1.6 Note: For any two non empty subsets A and B of the lattice L, we define, A + B = < {a + b / a A, b B}>, AB = < { ab / a A, b B} >. That is A + B and AB are the convex sublattices of L, generated by the elements a + b and ab respectively. 1.7 Definition : A convex sublattice S of a lattice L, is called standard sublattice, if I < S, K > = < IS, IK > and I + < S, K >= < I + S, I + K >, hold for any pair{ I, K } of convex sublattice of L, whenever, neither S K nor I < S, K >, are empty. 1.8 Note : For the convex sublattices A and B of the lattice L, the equalities A (B] = ( A] (B] and < A, (B] > = (A] + (B] hold, where (X] denotes the ideal generated by X. 95

3 1.9 Definition: A convex sublattice D, of a lattice L, is called distributive sublattice, if < D, XY > = < D, X > < D, Y > and < D, X + Y > = < D, X > + < D, Y > hold, for any pair {X, Y} of convex sublattices of L, whenever, neither D X nor D Y are empty Theorem: For each d in L, {d} is a distributive sublattice of L. Proof: Let D = {d} L, where L is a lattice and D is a sublattice of lattice L. Let {X, Y} be a pair of convex sublattices of L, such that, neither D X nor D Y are empty. Then, for D X ϕ and D Y ϕ, d X and d Y for d D. Therefore, < D, X > = X and < D, Y > = Y. < D, XY> = XY = < D, X > < D, Y > and < D, X + Y > = X + Y = < D, X > + < D, Y >, where D X ϕ and D Y ϕ. Hence D, is distributive sublattice of L Theorem:Every ideal and dual ideal of a lattice L is Convex sublattice of L.Conversely, every convex sublattice of L is the set intersection of an ideal and of a dual ideal. Proof: Let L be a lattice, if I is an ideal of L, clearly I is a sublattice of L and for any a,b in I with a b, [a,b] (b] I, hence I is a Convex sublattice of L. Similarly, every dual ideal of L is a convex sublattice of L. Conversely, suppose that C is a convex sublattice of L. Put I = { x L / x a for some a in C} and D = { x L / a x for some a in C }. 96

4 Let x, y I, then x a and y b for some a, b in C. And hence x y a b C. So that x y I. Also if y x a C then y a C and hence y I. Therefore I is an ideal. Similarly D is a dual ideal of L. Now we shall prove that C = I D. Clearly C I and C D and hence C I D. On the other hand, x I D, implies x a and b x for some a, b in C. Implies b x a and a, b C. Implies x [b,a] C, implies x C. Hence C = I D. Hence, we can say that a sublattice of a lattice L is convex if and only if it is the set intersection of a ideal and a dual ideal of L Theorem : An ideal D of lattice L, is distributive, if and only if, it is a distributive sublattice of L. Proof :Suppose an ideal D of lattice L is distributive sublattice. Then, for { X, Y } a pair of Convex sublattices of L we have <D, X+Y > = <D, X> + <D, Y> and <D,XY> = <D,X > <D,Y > holds, whenever, neither D X ϕ nor D Y ϕ. Since we have <X,Y> = X+Y for the ideals X,Y in L. Now D + XY = <D, XY> = <D,X> <D,Y> = (D+X) (D+Y), and 97

5 D+( X + Y) = < D, X +Y > = < D,X > + < D,Y > = (D + X) + (D + Y). The first equality gives precisely that D is distributive ideal. Conversely, D is distributive ideal of L. Using the obvious equality (XY] = (X] (Y] valid for any X, Y of L. < D, XY > = D + (XY] = D + (X] (Y] = ( D + (X] ) ( D + (Y] ), and < D, X > < D, Y > = ( D + (X] ( D + (Y] ). Therefore, < D, XY > = < D, X > < D, Y >. For every ideal D of L, we have D < D, X > and D < D, Y > D < D, X > + < D, Y >. Also X < D, X > and Y < D, Y > X + Y < D, X > + < D,Y>. Hence < D, X + Y > <D, X> + < D, Y > (1) Now < D, X > = D + (X] and < D, Y> = D + (Y]; < D, X + Y> = D + ( X + Y]. Clearly < D, X > + < D, Y > is a convex sublattice generated by the elements of the form ( d 1 + x 1 ) + ( d 2 + y 1 ), where d 1, d 2 D and x 1 x for x X, y 1 y for y Y. Since D < D, X + Y > and X + Y < D, X + Y > D + ( X+ Y ) < D, X + Y > (d 1 + d 2 ) + (x 1 + y 1 ) D + ( X + Y ) < D, X + Y > (d 1 + x 1 ) + (d 2 + y 1 ) < D, X + Y > < D, X > + < D, Y > < D, X + Y > (2) 98

6 From (1) and (2) we have < D, X + Y > = < D, X > + < D, Y >. Therefore, D is distributive sublattice of L Theorem : Let D be a convex sublattice of L. If x, y in L such that x + t = y + t, xy = ys, for some s, t in D, then < D, {x} > = < D, {y}>. Proof : Let D be a convex sublattice of L. Let x, y L, such that x + t = y + t, xs = ys, for some s,t in D. Clearly x + t <D,{x}>, for t D, x {x} and xs <D,{x}>, for s D, x {x} y + t < D, {x} > and ys <D,{x}>. Since, <D, {x}> is a convex sublattice, [ys, y+t] <D, {x}>. {y / ys y y +t} <D,{x}> y <D,{x}>. Since, D <D,{x}> and y <D,{x}>, we have <D,{y}> <D,{x}>. Similarly, for y+t < D, {y} > and ys < D, {y} >, we have x+ t < D, {y} > and xs < D, {y} >. Since < D, {y} > is a convex sublattice [xs,x+t] < D,{y}>. { x/xs x x+t } <D,{y}>. x <D,{y}>. Thus, we have < D,{x} > < D,{y} >. Therefore, <D,{x}> = <D,{y}>. 99

7 1.14 Theorem: Let L be a lattice and D be a distributive sublattice of L. If D satisfies (i) <D, XY> = <D,X> < D,Y> and (ii) <D, X + Y> = <D,X> + <D, Y> for all single element convex sublattices X, Y of L, then the binary relation θ D on L, defined as follows x y (θ D ) if and only if xy + t = x + y + t, xys = (x + y)s, for some s,t in D is a congruence relation. Proof : Assume that, D is a distributive sublattice of L, and the relation θ D on L, defined by x y (θ D ), if and only if, x + t = y + t, xs = ys, for some s, t D, with s t. To prove that θ D is a congruence relation, we verify the following : (i) x y (θ D ) if and only if xy x + y (θ D ). (ii) x y z, x y (θ D ) and y z (θ D ) x z (θ D ). (iii) x y and x y (θ D ) xz y z(θ) and x + z y + z (θ D ), for every z in L. Suppose x y (θ D ). xy + t = x + y + t, xys = (x + y) s ( as θ D is given in statement) xy (x + y) θ D. Suppose x y z, x y (θ D ) and y z (θ D ), x + t 1 = y + t 1, xs 1 = ys 1, for some s 1, t 1 D, with s 1 t 1 and y + t 2 = z + t 2, ys 2 = zs 2 for some s 2, t 2 D with s 2 t 2. Now x + t 1 + t 2 = y + t 1 + t 2 = z + t 1 + t 2 x + t 1 + t 2 = z + t 1 + t 2 x + t 3 = z + t 3, where t 3 = t 1 + t 2 and xs 1 = ys 1 xs 1 s 2 = ys 1 s

8 xs 3 = zs 3, where s 3 = s 1 s 2. Therefore x + t 3 = z + t 3 ; xs 3 = zs 3 x z(θ D ). Suppose x y and x y (θ D ). x + t = y + t, xs = ys for s t in D Now xz + t <D, XZ > = <D,X> <D, Z> = <D, {x}> <D,Z>, for X = {x} = <D, {y}> <D,Z>, since <D,{x}> = <D,{y}> = <D,Y> <D,Z> = <D,YZ>. Therefore, xz + t <D,YZ>. Hence xz + t yz + t for t D. Similarly, yz + t xz + t xz + t = yz + t. Now (xz)s = (xs)z = (ys)z = (yz)s. Therefore, xz + t = yz + t; (xz)s = (yz)s xz yz (θ D ) for every z in L. Similarly, x + z + t =x + t + z =y + t + z= y +z + t Therefore x + z + t = y + z + t. Also (x+z) s = xs + zs = ys + zs = (y+z)s Implies (x+z)s = (y+z)s. Therefore x + z y + z (θ D ) for every z, in L. Thus (θ D ) is a congruence relation. 101

9 1.15 Theorem: The above theorem can also be proved conversely as the relation θ D is defined in above theorem. That is, if D is a convex sublattice of L, such that the relation (θ D ) as defined in the above theorem is a congruence relation, then D is a distributive sublattice of L. Proof: Define a relation θ D as follows x y (θ D ) xy + t = x + y + t ; xys = (x + y) s for some s, t D Given that θ D is a congruence relation To show that D is distributive sublattice of L that is < D, XY> = <D, X> < D, Y> < D, X + Y> = <D, X> + < D, Y> when ever D X ϕ and D Y ϕ For any pair of convex sublattice {X, Y} of L. Clearly D <D, X> and D <D, Y> then D <D, X> <D, Y> Similarly X <D, X>, Y <D,Y> then XY <D, X> <D,Y> Therefore <D, XY> <D,X> <D,Y> Now, To prove that <D,X> <D,Y> <D,XY> Let P <D,X> <D,Y> implies P = st for s <D,X> and t <D,Y> such that d 0 x 0 s d 1 + x 1 d 4 + x 1 d 2 y 0 t d 3 + y 1 d 4 + y 1 where d 4 = d 1 + d 3 d 0 x 0 d 2 y 0 st (d 4 + x 1 ) (d 4 + y 1 ) d x 0 y 0 st (d 4 + x 1 ) (d 4 + y 1 ) where d = d 0 d 2 Take x 2 = x 1 + x, y 2 = y 1 + y where x D Y, y D Y So, we have d x 0 y 0 st (d 4 + x 2 ) (d 4 + y 2 ) 102

10 We have x 2 = x 1 + x for x D X Now, d 4 + x 2 + d 4 = d 4 + x 1 + x 2 + d 4 (since x 1 x 2 x 1 x 2 = x 2 ) = x 1 + x 2 + d 4 = x 2 + d 4 x 2 + d 4 = d 4 + x 2 + d 4 x 2 = d 4 + x 2 and x 2 x = (d 4 + x 2 ) x for some d 4, x in L Similarly y 2 (d 4 + y 2 ) θ D Therefore x 2 y 2 (d 4 + x 2 ) (d 4 + y 2 ) (θ D ) implies that there exists d in D such that d + x 2 y 2 = (d 4 + x 2 ) (d 4 + y 2 ) + d <D,XY> = <D,X> + <D,Y> follows dually Hence D is a distributive sublattice of L. Conclusion: In this chapter, we give a generalization of distributive ideals for convex sublattice called distributive covex sublattice. It is provided many important properties of Distributive ideals are also valid for Distributive Sublattice. We established the connections of Distributive Convex sublattices with Congruence classes of Lattices. 103

11 Research papers published relating to study under taken in this Dissertation. (1)Venkateswara Rao.J,Sree rama ravikumar.e, Modular and Classic ideals in directed below Joinsemilattices, International Journal of Algebra,vol.4,2010,no.18, (To appear in 2010) (2)Venkateswara Rao.J and Sree rama ravikumar.e.-weakly distributive and * semilattices, Asian Journal of Algebra 3(2):36-42,2010. (3)Venkateswara Rao.J and Sree rama ravikumar.e. Characterization of Standard and Distributive ideals in Semilattices, International Journal of Systemics, Cybernetics and Informatics, (ISSN ), Oct (1) Pp (4)Venkateswara Rao.J and Sree rama ravikumar.e., Distributive Convex sublattice, Southeast Asian Bulletin of Mathematics, January 2010,Vol.34(Number 1) pp (5)Venkateswara Rao.J and Sree rama ravikumar.e, Characterization of Supermodular semilattice, Southeast Asian Bulletin of Mathematics, 2010, Vol.34, pp.1-6 ( To appear in as per gallery proof). Research papers presented in International Conferences relating to the study undertaken in this Dissertation. (1)Venkateswara Rao.J and Sree rama ravikumar.e., Modular and Classic ideals in directed below Join semilattice, presented in the International Conference on Mathematical and Computer Models,2009, organized by P.S.G.College of Technology, Coimbatore and published in proceedings by Narosa publishers, ISBN ,pp (2)Venkateswara Rao.J and Sree rama ravikumar.e., Characterization of Standard and Distributive ideals in semilattice, published in Proceedings of the International 104

12 Conference on Systemics,Cybernetics and Informatics, organized by Penta gram Research Center,Pvt.Ltd., Hyderabad,January 2010,pp