On Finitely Generated n-ldeals of a Nearlattice

Abstract

The thesis studies extensively the nature of finitely generated n-ideals of a near lattice. Many authors including [1], [34] and [35] have studied about n-ideals of a lattice. A convex sublattice containing a fixed element n of a lattice L, is called an n-idea/. If L has 'O', then replacing n by 'O', an n-ideal becomes an ideal. Similarly, if L has 1, an n-ideal becomes a filter replacing n by 1. Thus, the idea of n-ideals is a kind of generalization of both ideals and filters of lattices. Many authors including [27], [40], [47] and [56] done some work on n-ideals of a near lattice. A near lattice is a meet semilattice with the property that any two elements possessing a common upper bound, have a supremum. For two n-ideals I and J of a near lattice S, [27] has given a neat description of Iv J, while the set theoretic intersection is the Infinimum. So, I"(S), the set of all n-ideals of a near lattice S, is a lattice. An n-idea generated by a finite number of elements ai, a2, ...., are is called a. finitely generated n-deal and denoted by <a1, a2, ...., a,>"' while the n-ideal generated by a single lament x is called a principal n-ideal, denoted by <x> n· F n(S) and P11(S) denotes the set of all finitely generated n-ideals and the set of all principal n-ideals respectively. When n e S is medial and standard, then P 0(S) is meet semilattice. Moreover, when n is sesquimedial and neutral, then P n(S) is again a near lattice. In general F n(S) is a join semilattice. But when S is distributive, then Fn(S) is also a distributive lattice. In this thesis we prove several results on finitely generated n-ideals of a distributive near lattice S when n is an upper element. These results certainly extend and_ generalize many results on disjunctive, generalized Boolean, generalized Stone and relatively Stone near lattices. Since for a distributive near lattice S with n e S, in(S) is a distributive algebraic lattice, so it is pseudo complemented.