International Journal of Scientific Engineering and Technology (ISSN : 2277-1581 Volume No.2, Issue No.12, pp : 1221-1222 1 Dec. 2013 IJSET@2013 Page 1221 Study on Prime n-ideals of a Lattice 1 Md. Ekramul Islam, 2 Arjuman Ara, 3 Md. Rezwan Ahamed Fahim, 4 Md.Hafizur Rahman 1,3,4 Department of Mathematics,Pabna University of Science and Technology,Bangladesh. 2 Department of Textile Engineering,City University,Bonani,Dhaka-1213,Bangladesh. Corresponding Email : ekramul.mathru@gmail.com Abstract— For a fixed element n in a lattice L , any convex sub-lattice containing n is called an n –ideal.We generalize several results of prime ideals of a lattice in terms of n-ideals. We introduce the notion of relative prime n-ideals in a lattice and include some interesting results on this. Keywords—n-ideals, Prime n- ideals, Convex sub-lattice, distributive lattice. I. Introduction The idea of n-ideals in a lattice was first introduced by Cornish and Noor in several Papers vii, viii, ix, x,xi. For a fixed element n of a lattice L, a convex sub lattice containing n is called an n- ideal. If L has a ‘0’ then replacing n by 0, an n-ideals becomes an ideal. Moreover, if L has a ‘1’, an n-ideals becomes a filter by replacing n by ‘1’. Thus the idea of n-ideals is a kind of generalization of both ideals and filters of a lattice. So any results involving n-ideals will give a generalization of the results on ideals and filters with 0 and 1 respectively in a lattice. The set of all n-ideals of a lattice L is denoted by I n (L), which is an algebraic lattice under set-inclusion. Moreover, {L} and L are respectively the smallest and largest elements of I n (L), while the set–theoretic intersection is the infimum. II. Material and Methodology For any two n-ideals I and J of L, it is easy to check that IJ=IJ={x: x= m(i, n, j) for some iI, jJ} Where m(x,y,z)=(xy)(yz)(zx) and IJ = {x: i 1 j 1 ≤ x ≤ i 2 j 2, for some i 1 ,i 2 I and j 1 ,j 2 J}. The n-ideal generated by a 1 , a 2 , ……. a m is denoted by  a 1 , a 2 , …….a m  n . Clearly  a 1 , a 2 , …….a m  n =a 1  n  a 2  n  …….a m  n . The n- ideal generated by a finite number of elements is called a finitely generated n-ideal. The set of all finitely generated n-ideal is denoted by F n (L). Also the n-ideal generated by a single element is called a principal n-ideal. The set of all principal n-ideal of L is denoted by P n (L). We have a n = {xL: an ≤ x ≤an}. The median operation m(x, y, z)=(xy)(yz)(zx) is very well known in lattice theory. An n-ideal P of a lattice L is called prime if m(x, n, y) P;x,yL implies either xP or yP. An element s of a lattice L is called standard if for all x, yL, x(ys) = (xy)(xs). An element nL is called neutral if it is standard and for all x, yL, n(xy) = (nx)(ny). Of course 0 and 1 of a lattice are always neutral. An element nL is called central if it is neutral and complimented in each interval containing n. A lattice L with 0 is called sectionally complimented if [0,x] is complimented for all xL. A sub-lattice H of a lattice L is called a convex sub-lattice if for any a,b  H, a< c < b implies c  H. Example: in the fig below(fig-1) {0,a,c},{0,b,c} are convex lattices. III. Results and Tables Theorem 3.1: If P is a prime n-ideal of a lattice, then for any xL, at least one of x  n and x  n is a member of P. Proof: Observe that m(x  n, n, x  n) = n P. Thus either xn  P or xn  P. Theorem 3.2: If P is a prime n-ideal of a lattice, then P contains either (n] or [n), but not both. Proof: Suppose P is prime and P ⊉ (n]. Then there exist r < n such that r  P. Now let s[n). Then m(r, n, s) = (rn)(ns)(sr) = rnr = nP implies that sP. That is, P[n). Similarly, if P ⊉ [n), then we can show P  (n]. Finally suppose that P contains both (n] and [n). Let t L. Then tn  P and tn P. Then by convexity of n-ideals t  P. This implies P=L, which is a contradiction to the primeness of P. Corollary 3.3: If P is a prime n-ideal of a lattice L, then there exists at least one xL such that both x  n and x  n do not belong to P.