Peter Schuster

TCS Journal 2023 Journal Article

Radical theory of Scott-open filters

• Daniel Misselbeck-Wessel
• Peter Schuster

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.

FLAP Journal 2018 Journal Article

Suzumura Consistency, an Alternative Approach.

• Peter Schuster
• Daniel Misselbeck-Wessel

Suzumura consistency is known as a sufficient and necessary condition for a binary relation to have an order extension. We advocate the use of equivalent but negation-free forms of Suzumura consistency and of the related notion of compatible extension. From a methodological perspective, our proposals make possible to work more abstractly, in the algebra of relations, and to give more direct proofs. To illustrate this we reconsider various forms and proofs of the order extension principle. As a complement we adopt to quasi-orders J.L. Bell’s argument that Gödel–Dummett logic is necessary for order extension.

FLAP Journal 2016 Journal Article

Editorial.

• Hannes Leitgeb
• Iosif Petrakis
• Peter Schuster
• Helmut Schwichtenberg

TCS Journal 2008 Journal Article

The Zariski spectrum as a formal geometry

• Peter Schuster

We choose formal topology to deal in a basic manner with the Zariski spectra of commutative rings and their structure sheaves. By casting prime and maximal ideals in a secondary role, we thus wish to prepare a constructive and predicative framework for abstract algebraic geometry. In contrast to the classical approach, neither points nor stalks need occur, let alone any instance of the axiom of choice. As compared with the topos-theoretic treatments that may be rendered predicative as well, the road we follow is built from more elementary material. The formal counterpart of the structure sheaf which we present first is our guiding example for a notion of a sheaf on a formal topology. We next define the category of formal geometries, a natural abstraction from that of locally ringed spaces. This allows us to eventually phrase and prove, still within the language of opens and sections, the universal property of the Zariski spectrum. Our version appears to be the only one that is explicitly point-free.

TCS Journal 1980 Journal Article

Multiplicative complexity of some rational functions

• Werner Hartmann
• Peter Schuster

Modifying Strassen's theorem on the substitution-method [2], we obtain sharp bounds for the multiplicative complexity of evaluating r-tuples of the following rational functions: products, continued fractions, compositions and quotients of polynomials in one variable with general coefficients.