Let E be a locally compact space which can be expressed as the union of an increasing sequence of compact subsets Kn (n =1, 2, …) and let μ be a positive Radon measure on E. Ω is the space of equivalence classes of locally integrable functions on E. We denote the equivalence class of a function f by and if is an equivalence class then f denotes any function belonging to f. Provided with the topology defined by the sequence of seminorms

Ω is a Fréchet space. The dual of Ω is the space φ of equivalence classes of measurable, p.p. bounded functions vanishing outside a compact subset of E. For a subset Γ of Ω, the collection Λ of all ∊Ω, such that for each g∊Γ the product fg is integrable, is called a Köthe space and Γ is said to be the denning set of Λ. The Köthe space Λx which has Λ as a denning set is called the associated Kothe space of Λ. Λ and Λx are put into duality by the bilinear form

This paper may be regarded as a continuation of the investigations begun in [2]; certain intrinsic lattice topologies are studied, especially the order and ideal topologies in Boolean algebras, bicompactly generated lattices, and other more general structures. The results of [1], [2], and [3] are shown to be closely related. It is proved that the ideal topology on any Boolean algebra has a closed subbase consisting of all sublattices, whereas the order topology on an atomic Boolean algebra has a closed subbase consisting of all sub-complete lattices. It is also shown that the order topology on an atomic Boolean algebra is autouniformizable (in the sense defined by Rema [3]) and, if the ground set is infinite, strictly coarser than the ideal topology. The conditions Cl and C3 on a lattice, introduced by Kent [1], are shown to be slightly stronger than the condition “ bicompactly generated ”, and in complete lattices, where these conditions are satisfied, the order topology is shown to be coarser than the ideal topology.

This paper is a continuation of [1]. We begin with the notations for the sequence spaces considered in this paper. Let Γ be the space of sequences x = {xp} of complex numbers such that |xp|1/p⃗0 as p⃗∞. Γ can also be regarded as the space of integral functions f(z) = . The sequence space Γ is a vector space over the complex numbers with seminorms

In this paper we prove the following:

Let S be a regular semigroup anda, bany elements of S such that Jb = ≦ Ja. Then, for each idempotent e∈Ja, there exists an idempotent f∈ Jb such that f = ≦e.