Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

Abstract

$W$ is the category of archimedean $l$-groups with distinguished weak order unit, with $l$-group homomorphisms which preserve unit. This category includes all rings of continuous functions $C(X)$ and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in $W$. There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each $W$– object$B$, we define a topology $\tau^{B}$ and a convergence $\mathop{\longrightarrow}\limits^{B}$. These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject $A$ of $B$ is dense iff $A$ is epically embedded. The topology is $T_{1}$, but only sometimes Hausdorff or an $l$-group topology. The convergence is a Hausdorff $l$-group convergence, but only sometimes topological. The associations of $B$ to $\tau^{B}$, and to $\mathop{\longrightarrow}\limits^{B}$, are functorial.