In this paper, we tailor-make new approximation operators inspired by rough set theory and specially suited for domain theory. Our approximation operators offer a fresh perspective to existing concepts and results in domain theory, but also reveal ways to establishing novel domain-theoretic results. For instance, (1) the well-known interpolation property of the way-below relation on a continuous poset is equivalent to the idempotence of a certain set-operator; (2) the continuity of a poset can be characterized by the coincidence of the Scott closure operator and the upper approximation operator induced by the way below relation; (3) meet-continuity can be established from a certain property of the topological closure operator. Additionally, we show how, to each approximating relation, an associated order-compatible topology can be defined in such a way that for the case of a continuous poset the topology associated to the way-below relation is exactly the Scott topology. A preliminary investigation is carried out on this new topology.

Abstract:In this paper, the˛waybelow relation, which is determined by O 2 -convergence, is characterized by the order on a poset, and a sufficient and necessary condition for O 2 -convergence to be topological is obtained.

In the present paper, we mainly focus on the symmetry of the solutions of a given PDE via Lie group method. Meanwhile we transfer the given PDE to ODEs by making use of similarity reductions. Furthermore, it is shown that the given PDE is self-adjoining, and we also study the conservation law via multiplier approach.

The notions of o s {o}_{s} -convergence and S ∗ {S}^{\ast } -doubly quasicontinuous posets are introduced, which can be viewed as common generalizations of Birkhoff’s order-convergence and S ∗ {S}^{\ast } -doubly continuous posets, respectively. We first consider the relationship between o s {o}_{s} -convergence and B-topology and show that the topology induced by o s {o}_{s} -convergence according to the standard topological approach is the B-topology precisely. Then, the topological characterization for the S ∗ {S}^{\ast } -doubly quasicontinuity is presented. It is proved that a poset is S ∗ {S}^{\ast } -doubly quasicontinuous iff the poset equipped with the B-topology is locally hyperclosed iff the lattice of all B-open subsets of the poset is hypercontinuous. Finally, the order theoretical condition for the o s {o}_{s} -convergence being topological is given and the complete regularity of B-topology on S ∗ {S}^{\ast } -doubly quasicontinuous posets is explored.

In this paper, we introduce the notion of super finitely separating functions which gives a characterization of RB-domains. Then we prove that FS-domains and RB-domains are equivalent in some special cases by the following three claims: a dcpo is an RB-domain if and only if there exists an approximate identity for it consisting of super finitely separating functions; a consistent join-semilattice is an FS-domain if and only if it is an RB-domain; an L-domain is an FS-domain if and only if it is an RB-domain. These results are expected to provide useful hints to the open problem of whether FS-domains are identical with RB-domains.