In this paper some results on direct summands of Goldie extending elements are studied in a modular lattice. An element of a lattice with is said to be a Goldie extending element if and only if for every there exists a direct summand of such that is essential in both and . Some characterizations of decomposition of a Goldie extending element in a modular lattice are obtained.
In this paper we generalize a result of Libkin concerning direct product decompositions of lattices.
Let be the space of continuous real-valued functions on X, with the topology of pointwise convergence. We consider the following three properties of a space X: (a) is Scott-domain representable; (b) is domain representable; (c) X is discrete. We show that those three properties are mutually equivalent in any normal T₁-space, and that properties (a) and (c) are equivalent in any completely regular pseudo-normal space. For normal spaces, this generalizes the recent result of Tkachuk that is...
We study domain-representable spaces, i.e., spaces that can be represented as the space of maximal elements of some continuous directed-complete partial order (= domain) with the Scott topology. We show that the Michael and Sorgenfrey lines are of this type, as is any subspace of any space of ordinals. We show that any completely regular space is a closed subset of some domain-representable space, and that if X is domain-representable, then so is any -subspace of X. It follows that any Čech-complete...