Radical theory of Scott-open filters

Abstract

Following the Kronecker–Duval or D5 philosophy of dynamic evaluation in computer algebra, the dynamical proof method was brought into constructive algebra in order to obtain computational interpretations of the individual, concrete instances of the Kuratowski–Zorn Lemma modern algebra abounds with. We now push this approach into complete lattices, and thus capture the computational content also of generic, abstract forms of the axiom of choice such as the Teichmüller–Tukey lemma.To this end we first strengthen the (key lemma for) the Hofmann–Mislove theorem about Scott-open filters (SOF): to be a member of a SOF is equivalent, with the axiom of choice, to membership of all complete extensions. This then allows for a constructive treatment: membership of the given SOF means that from the potential element one can grow a finite labelled binary tree of an inductively defined type such that every branch of the tree eventually dips into the SOF.The ideal objects characteristic for invocations of the Kuratowski–Zorn Lemma are thus approximated and actually replaced by paths in those trees, by which we shed light on the interaction of transfinite methods and their dynamical interpretation. The first test case includes abstract dependence, especially bases of vector spaces.Key to the above lies in taking Erné's saturation map as a radical operator for Scott-open sets. Through a generalised inductive definition this radical can be pinned down in terms of universal properties; and reaches into classical pointfree topology, as exemplified by a succinct proof of Isbell's spatiality theorem.