Standard Ideal and Filter of a Lattice

Abstract

This thesis studics the nature of Standard ideal of a lattice. The idea of standard ideal in lattice was first introduced by G. Gratzcr and E.T. Schmidt. The characterization of standard ideal was lirst introduced by M. F. Jamowitz. It had extended the ideal to convex sub lattices and proved many result of homomorphism by E. Fried and E.T. Schmidt. First we can define inlimum of two ideals of a lattice in their set theoretic intersection but supremum of two ideals I and J. In a lattice L is given by I ˅ J = ( x ϵ L : x I ˅ J , I ˅ J for some I ϵ I,j ϵ J). In a distributive lattice, two ideals 1 and J, the supremum i.e., 1 v I = { i ˅ j:i ϵ l, j ϵ J, where i, j exists}. But in a general lattice the formula for the supremum of two ideals is not easy. We start in chapter one the lemmas which gives the formula for the supremum of two ideals. An ideal I of a latticeL is called standard if and only if 1 is standard as all element of 1(L) the lattice of all ideals of L. That is of any ideals I,Lϵ I(L),1˄ (I v S) = (I˄ L) v (I˄ S) Any element of a lattice is standard if and only if it is distributive and modular. Thus, in a modular lattice every distributive element is standard. Not only that in a modular lattice every standard element is also neutral. Therefore, all ideal is standard if and only if it is both distributive and modular. Since a neutral element n of L is modular if and only if I (L) is modular. So every distributive ideal ofL is standard when L is modular and n is neutral. A congruence of a lattice L is called standard if for some standard ideal SoiL. A meet semi lattice together with the properly that any two elements possessing a common upper bound have a suprimum. For any two lattice L1 and L2 1 a map φ: L1 —> L2 is called an isotone if for x, y ϵ L any with x ≤ y implies φ(x) ≤ φ(y), also the above mapping is called a meet homomorphism if for all x, y ϵ L .φ(x ˄ y) = φ(x) ˄ φ(y). Therefore, meet homomorphism is all isotone and φ(x) v φ(y) ≤φ(x v y). Therefore, φ(x) v φ(y) exist by upper bound property of L2 . Chinthayamma Malliah and Parameshwara Bhatta have characterize those lattices, whose all congruence are standard and neutral. Here we generalize characterization of those lattice whose all congruence are standard. In this thesis, we have studied several properties of Standard ideal of a lattice. Moreover, we give several results on Standard ideal of a lattice which certainly extend and generalize many results in lattice theory.

In Chapter two, we have discussed ideals, congruence, length and covering conditions, For any subset K of a lattice L, (K] denotes the ideal generated by K. Infiniuni of two ideals of a lattice is their set theoretic intersection. supremum of two kals I and J in a lattice L is given by I V J= I ˅ J = {X ϵ L I X :≤ i ˅ j for some iϵ 1,j ϵ J). Corn ish and Hickman in [3] showed that in a distributive lattice L for two ideals I and J, I v J = {i v j : i ϵ 1,j ϵ J, where Iv / exists}. But in a general lattice the formula for the suprernum of two ideals is not very easy. Which are explain with some examples and generalized many theorems of them. In Chapter three, Standard and Neutral elements of a lattice and Traces have been discussed. Standard elements in lattices were first studied in depth by, Gratzer and schmidt [15]. Since then little attenton has been paid to these notions. A lower Semi lattice is said to have the upper bound property if the suprcnium of any two elements automatically exists when they share a common upper bound. According to Gratzer and 4 Schmidt [15] ifa is an element of a lattice L then, (i) a is called distributive if a ˅ (r ˄ s) = (a ˅ r) ˄ (a ˅ s) for all r, s ϵ L; (ii) a is called standard if A (s ˅ a) = (r ˄ s) V (r ˄ a) fir all r, s ϵ L; (iii) a is called neutral if the sub lattice generated by r, s and a is distributive for all i, j ϵ L i.e., (a ˄ r)v(r ˄ s)v(s ˄ a) ˄ (a v r) ˄ (r ˄ s) ˄ (s v c )for all r, s ϵ L. Standard and Neutral elements are essential for the further development of standard ideals. In chapter four we give a description of Prime ideals, minimal prime ideals and normal. We have also studied Minimal prime n- ideals of' a lattice, we give some characterizations on minimal prime n-ideals which are essential for the further development of this chapter. Here we provide a number of results which are generalizations of the results on Normal and generalized Normal lattices. In chapter five we studied relatively pseudocomplemented of a lattice. We have also studied Multiplier extentions of pseudocomplemented lattices. These have been studied by Cornish and l-Iicman [3] and many other authors. I lere we extend several results of Cornish and Hicman to lattices. Pseudocompleniented distributive lattices satisfying Lce s identities l'orm educational subclasses denoted by Bn, 1 ≤ n≤ ω. Cornish and Mandelker have studied distributive lattices analogues to B1 -lattices and relatively B-lauices. Moreover, Cornish. I3cazer and Davey have idependently obtained several characterizations of sectionally Bm lattices and relatively Bm lattices. These have been studied by Cornish and Hicman and many other authors. Here we extend several results of Cornish and Hicrnan to lattices.

Chapter six introduces the concept of standard ideals, homomorphism, kernels, which have been studied by Gratzer, Schmidt and many other authors. We have given a characterization of standard ideals also characterize in a lattice every standard ideal in a homomorphism kernel of at least one congruence relation. Noor [32] has introduced the concept of standard n- ideals of a lattice. We conclude this thesis with some more properties of standard and neutral ideals, which are the basic concept or this thesis.