Study of Principal n-ideals of a Lattice

Abstract

This thesis studies extensively the Principal n-ideals of a lattice. The idea of n-ideals in a lattice was first introduced by Cornish and Noor in studying the kernels around a particular element n, of a skeletal congruence on a distributive lattice. Then Latif and Ayub Ali in their thesis studied thoroughly on the n-ideals and established many valuable results. For a fixed element n of a lattice L, a convex sublattice of L containing n is called an n-ideal. If L has a "0", then replacing n by 0, an n-ideal becomes an ideal and if L has a "1" then it becomes a filter by replacing n by I. Thus, the idea of n-ideals is a kind of generalization of both ideals and filters of lattices. The n-ideal generated by a finite number of elements of a lattice is called a finitely generated n-ideal, while the n-ideal generated by a single element is known as a principal n-ideal. Latif in his thesis has given a neat description on finitely generated n-ideals of a lattice and has provided a number of important results on them. For a lattice L, the lattice of all n-ideals of L and the lattice of all finitely generated n-ideals of L are denoted by In (L) and Fn (L) respectively, while Pn (L) represents the set of principal n-ideals of L. In this thesis, we devote ourselves in studying several properties on Pn (L) and Fn (L) which will certainly enrich many branches of lattice theory. Our results in this thesis generalize many results on normal, relatively normal, m-normal and relatively m-normal lattices. We also introduce the concept of n-annulets and α -n-ideal in studying Pn (L). In this connection it should be mentioned that if L has a 0, then putting n = 0 we find that Fn (L) is the set of all principal ideals of L which is isomorphic to L. Thus, for every result on Fn (L) in this thesis, we can obtain a result for the lattice L with 0 by substituting n = 0. Hence the result in each chapter of the thesis regarding Fn (L) are generalizations of the corresponding results in lattice theory. In chapter 2, we discuss some fundamental properties of n-ideals, which are basic to this thesis. Here we give an explicit description of Fn (L) and Pn (L) which are essential for the development of the thesis. Though Fn (L) is always a lattice, Pn (L) is not even a semilattice. But when n is a neutral element, Pn (L) becomes a meet semilattice. Moreover, we show that Pn (L) is a lattice if and only if n is a central element, and then in fact, Pn (L) = Fn (L). We also show that, for a neutral element n, the lattice L is complemented if and only if Pn (L) is so. In this chapter we also discuss on prime n-ideals. We give several properties and characterizations of prime n-ideals. We include a proof of the generalization of Stone's separation theorem. We also include a new proof of the result that for a distributive lattice L, Fn (L) is generalized Boolean if and only if prime n-ideals are unorderd. Chapter 3 discusses on 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 lattices. We prove that for a distributive lattice L, Fn (L) is normal if and only if each prime n-ideal of L contains a unique minimal prime n-ideal. We also show that if n is central in L, then Pn (L) is a normal lattice if and only if any two minimal prime n-ideal are comaximal which is also equivalent to < x > n ∩ <y> n = {n} implies <x> n* v <y> n*=L.

In chapter 4 we introduce the notion of relative n-annihilators <a, b >n. We characterize distributive and modular lattices in terms of relative n-annihilators. Then we generalize several results of Mandelker on annihiltors. We use these to characterize those Fn (L) which are relatively normal lattices. Among many results we have shown that for a central element n, Pn (L) is a relatively normal lattice, if and only if any two incomparable prime n-ideal are comaximal . What is more, this is also equivalent to the condition <<a >n,< b >n> v <<b >n,< a >n> = L for all a,b ϵL. Pseudocomplemented distributive lattices satisfying Lee's identities form equational subclasses denoted by Bm , - 1 ≤ m ˂ w Cornish have studied distributive lattices analogues to Bm-lattices and relatively Bm-lattices. He referred then as m-normal lattices.Moreover, Beazer and Deavy have each independently obtained several characterizations of (sectionally) Bm -lattices and relatively Bm -lattices. In chapter 5 we generalize their results by studying finitely generated n-ideals which form a m-normal and a relatively m-normal lattice .We show that for a central element n ϵ L, Pn(L) is m-normal if and only if for any m+1 distinct minimal prime n-ideals P0 ............., Pn of L, P0 v ................v Pm = L. In this chapter we also show that for a central element n ϵ L, Pn (L) is relatively m-normal if and only if any m+1 pairwise incomparable prime n-ideals are comaximal. Chapter 6 introduces the concept of n-annulets and α -n-ideals of a lattice. Here we include several result on the set of n-annulets An(L) when n is a central element of L. We proved An(L) is relatively complemented if and only if Pn(L) is sectionally quasi-complemented.