Epimorphisms of separated superconvex spaces (Q1366307)

The characterization of the epimorphisms of an algebraic theory proves often to be a difficult problem, as classical examples e.g. in the theory of groups, semigroups or rings show. The author investigates this problem for so-called superconvex spaces, which are a canonical generalization of countably convex subsets of Hausdorff linear topological spaces, which were investigated by \textit{G. Jameson} [cf. ``Ordered linear spaces'', Lect. Notes Math. 141 (1970; Zbl 0196.13401)], who called them CS-compact sets. Denoting by \(\Omega_{sc} :=\{(\alpha_i \mid i\in \mathbb{N}) \mid\alpha_i \geq 0\), \(i\in \mathbb{N}\), \(\sum^\infty_{i=1} \alpha_i =1\}\) the set of superconvex or countably convex series, a set \(C\) is called a superconvex space, if there is a mapping \[ \cdot: \Omega_{sc} \times C^\mathbb{N}\to C \] denoted by a formal sum \((\widehat \alpha,c) \mapsto \sum^\infty_{i=1} \alpha_ic_i\), \(\widehat \alpha\in \Omega_{sc}\), \(c= (c_i \mid i\in \mathbb{N})\in C^\mathbb{N}\), such that the following equations are satisfied: (SC 1) \(\sum^\infty_{i=1} \delta_{ik} c_i=c_k\), (SC 2) \(\sum^\infty_{i=1} \alpha_i (\sum^\infty_{k=1} \beta_{ik} c_k)= \sum^\infty_{k=1} (\sum^\infty_{i=1} \alpha_i \beta_{i k}) c_k\), \((\alpha_i \mid i\in \mathbb{N})\), \((\beta_{ik} \mid k\in \mathbb{N})\in \Omega_{sc}\). A morphism of superconvex spaces \(f:C_1 \to C_2\) is a mapping preserving these operations, i.e. \[ f\left( \sum^\infty_{i=1} \alpha_ic_i \right)= \sum^\infty_{i=1} \alpha_i f(c_i), \] \(c_i\in C\), \(i\in \mathbb{N}\). Convex spaces and their morphisms are introduced analogously by substituting the set of convex operations \(\Omega_c: =\{(\alpha_1, \dots, \alpha_n) \mid n\in \mathbb{N}\) and \(\alpha_i \geq 0\), \(i\in \mathbb{N}\), \(\sum^n_{i=1} \alpha_i=1\}\) for \(\Omega_{sc}\) in the above definition. The author succeeds in giving a characterization of the epimorphisms for an important special type, namely the so-called separated superconvex spaces with the help of a closure operator (cf. Theorem 3.5 and 3.6), defined by convex operations. Actually the result is proved for convex, separated spaces in 3.5 and then carried over to the superconvex case. A corresponding result is also proved for preseparated superconvex spaces in 4.1, a generalization of separated superconvex spaces.