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On the Lattice of Convex Sublattices R.Subbarayan 1 and A.Vethamanickam 2 1Mount Zion College of Engineering and Technology, Pudukkottai, Tamilnadu, India 622 507

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On the Lattice of Convex Sublattices R.Subbarayan 1 and A.Vethamanickam 2 1Mount Zion College of Engineering and Technology, Pudukkottai, Tamilnadu, India 622 507 10471 R.Subbarayan et al./ Elixir Dis. Math. 50 (2012) 10471-10474 Available online at www.elixirpublishers.com (Elixir International Journal) Discrete Mathematics Elixir Dis. Math. 50 (2012) 10471-10474 On the Lattice of Convex Sublattices R.Subbarayan 1 and A.Vethamanickam 2 1Mount Zion College of Engineering and Technology, Pudukkottai, Tamilnadu, India 622 507. 2Department of Mathematics, Kamarajar Arts College, Surandai Tamilnadu, India. ARTICLE INFO ABSTRACT Article history: Let L be a finite lattice. A sublattice K of a lattice L is said to be convex if a,b K; c L, Received: 11 April 2012; Received in revised form: a c b imply that c K. Let CS(L) be the set of convex sublattices of L including the 5 September 2012; empty set. Then CS(L) partially ordered by inclusion relation, forms an atomic algebric Accepted: 15 September 2012; lattice. Let P be a finite graded poset. A finite graded poset P is said to be Eulerian if its M bius function assumes the value (x; y) = (-1) l(x;y) for all x y in P, where l(x, y) = (y)- Keywords (x) and is the rank function on P. In this paper, we prove that the lattice of convex Posets, Lattices, sublattices of a Boolean algebra B n, of rank n, CS(B n) with respect to the set inclusion Convex Sublattice, relation is a dual simplicial Eulerian lattice. But the structure of the lattice of convex Eulerian Lattices, sublattices of a non- Boolean Eulerian lattice with respect to the set inclusion relation is not Simplicial Eulerian Lattice. yet clear. © 2012 Elixir All rights reserved. Introduction An equivalent definition for an Eulerian poset is as follows: The study of lattice of convex sublattices of a lattice was Lemma 2.2 [8] A finite graded poset P is Eulerian if and only if started by K.M. Koh [3], in the year 1972. He investigated the all intervals [x, y] of length l 1 in P contain an equal number internal structure of a lattice L, in relation to CS(L), like so of elements of odd and even rank. many others for various algebraic structures such as groups, Example : Every Boolean algebra of rank n is Eulerian and the Boolean algebras, directed graphs and so on. lattice C of Figure 1 is an example for a non-modular Eulerian In [3], several basic properties of CS(L) have been studied 4 and proved that "If L is complemented then CS(L) is lattice. Also, every C n is Eulerian for n 4. complemented". Also, the connection of the structure of CS(L) with those of the ideal lattice I(L) and the dual ideal lattice D(L) are examined. K.M. Koh, derived the best lower bound and upper bound for the cardinality of CS(L), where L is finite. In a subsequent paper[1], C.C. Chen, and K.M. Koh, proved that CS(LxK) [(CS(L) - ) X (CS(K) - )] Finally, they proved that when L is a finite lattice and CS(L) CS(M) and if L is relatively complemented (complemented) then M is relatively complemented (complemented). This is true for Eulerian lattices, since an Eulerian lattice is relatively complemented. These results gave motivation for us to look into the connection between L and CS(L) for Eulerian lattices which Figure 1 are a class of lattices not defined by identities. Since a B , n Lemma 2.3 [10] If L and L are two Eulerian lattices then L X Boolean algebra of rank of n, for n = 1; 2…, is Eulerian, we start 1 2 1 L is also Eulerian. looking into the structure of CS(B ). 2 n We note that any interval of an Eulerian lattice is Eulerian In section 3, we prove that CS(B ), the lattice of convex n and an Eulerian lattice cannot contain a three element chain as sublattices of B with respect to the set inclusion relation is a n an interval. For more structures of Eulerian lattices, see [12] dual simplicial Eulerian lattice. But the structure of the lattice of convex sublattices of non-Boolean Eulerian lattice with respect Definition 2.4 A poset P is called Simplicial if for all t P, to the set inclusion relation is not yet clear. [0, t] is a Boolean algebra and P is called Dual Simplicial if for Preliminaries all t P, [t, 1] is a Boolean algebra. Throughout this section CS(L) is equipped with the partial The following remark is the example 1.1.17 in the book of order of set inclusion relation. R.P.Stanley [9] Definition 2.1 . A finite graded poset P is said to be Eulerian if its M bius function assumes the value (x, y) = (-1) l(x;y) for all Remark 2.5 = x y in P, where l(x, y) = (y)- (x) and is the rank function on P. Tele: E-mail addresses: [email protected] © 2012 Elixir All rights reserved 10472 R.Subbarayan et al./ Elixir Dis. Math. 50 (2012) 10471-10474 Theorem 2.6 The lattice of convex sublattices of a Boolean a3 = + + + algebra B n of rank n, CS(B n) with respect to the set inclusion relation is a dual simplicial Eulerian lattice . + … + Proof: It is clear that rank of CS(B n) is n + 1. First we prove that the interval [ B n] is Eulerian. Continuing like this, we get, That is to prove that this interval has same number of elements of odd and even rank. a4 = + + + Let a i be the number of elements of rank i in CS(B n). + … + Since the elements of rank 1 in CS(B n) are just the singleton n subsets of B n, we have, a1 = 2 . and so on The rank two elements in the interval [ B n] are the two- a element chains. n-2 = + + + We have to determine the total number of two-element chains in B n. Since there are atoms in B n , the number of an-1 = + + two-element chains containing 0 in B n is Since there are an = + edges emanating from an atom in B n there are Case(i) : Suppose n is even. two-element chains containing an atom in B n. a1 – a 2 + a 3 – … - a n-2 + a n-1 - a n = 2n – A rank two element is connected by edges to some of the rank three elements and since there are rank 2 elements in B n, we have the number of two-element chains from the rank two elements of B n are . Similarly, the total number of two-element chains from the rank three elements are . Considering all the elements upto rank n - 1 the total number of two element chains in B n is a2 = + + + + … + A rank three element in [ B n] is a sublattice B 2 of B n. There are rank two elements in B n. Therefore, the number of B 2's containing 0 is = 2n- [ There are atoms in B n . If a is an atom then [a, 1] [ Bn-1. From ‘a’ to a rank three element in [a, 1] we have a sublattice B 2 with a as the lowest element. + Since there are such rank three elements we have [ + the number of such B 2's is In all, the number of B 2's with an atom as the lowest element is Similarly, the number of B 2's with a rank two element as the lowest element is . Proceeding like this, we get, 10473 R.Subbarayan et al./ Elixir Dis. Math. 50 (2012) 10471-10474 The B 2's with a in the top is an upper inerval in [0, a]. n = 2 – - - Therefore, the number of such B 2's are exactly . - . - The top elements in the B 2's with a as a middle element are just the n-k atoms of the interval [a, 1] = 2n ... We observe that a B 2 with a in the middle has one of the atoms of [a, 1] as the top element and one of the co-atoms of [0; a] as the bottom element. = 2n ... Now, fix an atom in [a, 1]. Since there are k co-atoms in n = 2 [0, a] the number of B 2's with this atom as a top element and a in the middle is k. = 2n - 2n = 0 This is true for every atom in [a,1]. Since there are n - k Therefore, a – a + a – … - a + a - a = 0 1 2 3 n-2 n-1 n such atoms the total number of B 's having a in the middle is Case(i) : Suppose n is odd. 2 a1 – a 2 + a 3 – … - a n-2 + a n-1 - a n n The B 2's with a in the bottom is a lower interval in [a,1]. 2 -[ Therefore, the number of such B 2's are [ Hence the total number of rank 2 elements in [ Bn] is + The rank 3 elements in [ Bn] is isomorphic to B 3 [ + containing a. There are four possibilities, namely, a may be a top element or may be a rank 1 element or may be a rank 2 element or may be a bottom element of B 3. The B 's with a in the top is an upper interval in [0, a]. 3 Therefore, the number of such B 3's are exactly Suppose that a is a rank 2 element of the B 3's, that is, a co- atom of the B 's.
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