Some further results on pointfree convex geometry

Inspired by locale theory, pointfree convex geometry was first proposed and studied by Yoshihiro Maruyama. In this paper, we shall continue to his work and investigate the related topics on pointfree convex spaces. Concretely, the following results are obtained: (1) A Hofmann-Lawson-like duality for pointfree convex spaces is established. (2) The M-injective objects in the category of S-0-convex spaces are proved precisely to be sober convex spaces, where M is the class of strict maps of convex spaces; (3) A convex space X is sober iff there never exists a nontrivial identical embedding i : X hooked right arrow Y such that its dualization is an isomorphism, and a convex space X is S-D iff there never exists a nontrivial identical embedding k : Y hooked right arrow X such that its dualization is an isomorphism. (4) A dual adjunction between the category CLatD of continuous lattices with continuous D-homomorphisms and the category CSD of S-D-convex spaces with CP-maps is constructed, which can further induce a dual equivalence between CSD and a subcategory of CLat(D); (5) The relationship between the quotients of a continuous lattice L and the convex subspaces of cpt(L) is investigated and the collection Alg(Q(L)) of all algebraic quotients of L is proved to be an algebraic join-sub-complete lattice of Q(L) of all quotients of L, where cpt(L) denote the set of non-bottom compact elements of L. Furthermore, it is shown that Alg(Q(L)) is isomorphic to the collection Sob(P(cpt(L))) of all sober convex subspaces of cpt(L); (6) Several necessary and sufficient conditions for all convex subspaces of cpt(L) to be sober are presented.