Zhao, Dongsheng

A directed complete poset (dcpo, for short) P is said to satisfy the Lawson condition if the restriction of the Scott topology and the Lawson topology on the set of maximal points coincide. In this paper we investigate such dcpos, in particular their maximal point spaces. The main result is that for any dcpo satisfying the Lawson condition, the maximal point space is well- ltered and coherent (the intersection of any two saturated compact sets is compact). The relationship of such dcpos with other classes of dcpos are also considered.

We prove that for every T0 space X, there is a well-filtered space W(X) and a continuous mapping ηx : X⭢ W(X), such that for any well-filtered space Y and any continuous mapping 𝒇 : X⭢Y there is a unique continuous mapping 𝒇^: W(X)⭢Y such that 𝒇=𝒇^∘ηX. Such a space W(X) will be called the well-filterification of X. This result gives a positive answer to one of the major open problems on well-filtered spaces. Another result on well-filtered spaces we will prove is that the product of two well-filtered spaces is well-filtered.

We lift the notion of quasicontinuous posets to the topology context, called quasicontinuous spaces, and further study such spaces. The main results are: (1) A $T_{0}$ space $(X,\tau)$ is a quasicontinuous space if and only if $SI(X)$ is locally hypercompact if and only if $(\tau_{SI},\subseteq)$ is a hypercontinuous lattice; (2) a $T_{0}$ space $X$ is an $SI$-continuous space if and only if $X$ is a meet continuous and quasicontinuous space; (3) if a $C$-space $X$ is a well-filtered poset under its specialization order, then $X$ is a quasicontinuous space if and only if it is a quasicontinuous domain under the specialization order; (4) there exists an adjunction between the category of quasicontinuous domains and the category of quasicontinuous spaces which are well-filtered posets under their specialization orders.

Sobriety, well-filteredness, and monotone convergence are three of the most important properties of topological spaces extensively studied in domain theory. Some other weak forms of sobriety and well-filteredness have also been investigated by some authors. In this paper, we introduce the notion of Θ-fine spaces, which provides a unified approach to such properties. In addition, this general approach leads to the definitions of some new topological properties.

A nonempty compact saturated subset F of a topological space is called k-irreducible if it cannot be written as a union of two compact saturated proper subsets. A topological space is said to be co-sober if each of its k-irreducible compact saturated sets is the saturation of a point. Wen and Xu (2018) proved that Isbell's non-sober complete lattice equipped with the lower topology is sober but not co-sober. So far, it is still unknown whether every sober Scott space is co-sober. In this paper, we construct a dcpo whose Scott space is sober but not co-sober, which strengthens Wen and Xu's result.

In this paper we investigate -primary ideals which unify prime ideals and primary ideals. A number of main results about prime ideals and primary ideals are extended into this general framework. Prime ideals and primary ideals are two of the most important structures in commutative algebra. Although different from each other in many aspects, they share quite a number of similar properties as well( see [1] ). However, these two structures have been treated rather differently, and all of their properties were proved separately. It is therefore natural to examine whether it is possible to have a unified approach to studying these two structures. In this short paper we introduce the notion of -primary ideals where is a mapping that assigns to each ideal I an ideal (I) of the same ring. Such -primary ideals unify the prime and primary ideals under one frame. This approach clearly reveals how similar the two structures are and how they are related to each other. In the first section, we introduce ideal expansion and define primary ideals with respect to such an expansion. Besides the familiar expansions 0, 1 and B, we also have a new expansion M defined by means of maximal ideals. In the second section, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized primary ideals with respect to such expansions. In this paper, all the rings used are commutative rings with an multiplication identity and all the ring homomorphisms preserve the identity. We shall use Id(R) to denote the set of all ideals of the ring R.

Based on the topological Rudin's Lemma, we introduce the notions of Rudin set and well-filtered determined set in a topological space. Using such sets, we formulate and prove some new characterizations of well-filtered spaces and sober spaces. Part of the work was inspired by Xi and Lawson's work on well-filtered spaces. Our study also lead to the definition of a new class of spaces - the strong d-spaces, and some problems whose solutions will strengthen the understanding of the related structures.

A dcpo P is continuous if and only if the lattice C(P) of all Scott-closed subsets of P is completely distributive. However, in the case where P is a non-continuous dcpo, little is known about the order structure of C(P). In this paper, we study the order-theoretic properties of C(P) for general dcpo's P. The main results are: (i) every C(P) is C-continuous; (ii) a complete lattice L is isomorphic to C(P) for a complete semilattice P if and only if L is weak-stably C-algebraic; (iii) for any two complete semilattices P and Q, P and Q are isomorphic if and only if C(P) and C(Q) are isomorphic. In addition, we extend the function P 7! C(P) to a left adjoint functor from the category DCPO of dcpo's to the category CPAlg of C-prealgebraic lattices.

Keimel and Lawson (2009) proposed a set of conditions for proving the reflectivity of a category of topological spaces in the category of all T0 spaces. Recently, these conditions were used to prove the reflectivity of the category of all well-filtered spaces. We prove that, in certain sense, these conditions are not only sufficient but also necessary for a category of T0 spaces to be reflective. By applying this general result, we can easily deduce that several categories proposed in domain theory are not reflective, thereby answering a few open problems.