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 \(\mathcal {M}\)-injective objects in the category of \(S_0\)-convex spaces are proved precisely to be sober convex spaces, where \(\mathcal {M}\) is the class of strict maps of convex spaces; (3) A convex space X is sober iff there never exists a nontrivial identical embedding \(i:X\hookrightarrow Y\) such that its dualization is an isomorphism, and a convex space X is \(S_D\) iff there never exists a nontrivial identical embedding \(k:Y\hookrightarrow X\) such that its dualization is an isomorphism. (4) A dual adjunction between the category \(\textbf{CLat}_D\) of continuous lattices with continuous D-homomorphisms and the category \(\textbf{CS}_D\) of \(S_D\)-convex spaces with CP-maps is constructed, which can further induce a dual equivalence between \(\textbf{CS}_D\) and a subcategory of \(\textbf{CLat}_D\); (5) The relationship between the quotients of a continuous lattice L and the convex subspaces of \({\textbf {cpt}}(L)\) is investigated and the collection \({\textbf {Alg}}({\textbf {Q}}(L))\) of all algebraic quotients of L is proved to be an algebraic join-sub-complete lattice of \({\textbf {Q}}(L)\) of all quotients of L, where \({\textbf {cpt}}(L)\) denote the set of non-bottom compact elements of L. Furthermore, it is shown that \({\textbf {Alg}}({\textbf {Q}}(L))\) is isomorphic to the collection \({\textbf {Sob}}(\mathcal {P}({\textbf {cpt}}(L)))\) of all sober convex subspaces of \({\textbf {cpt}}(L)\); (6) Several necessary and sufficient conditions for all convex subspaces of \({\textbf {cpt}}(L)\) to be sober are presented.