On Scott Consequence Systems

Article type: Research Article

Authors: Dimov, Georgi | Vakarelov, Dimiter

Affiliations: Department of Mathematics and Computer Science, University of Sofia, Blvd. J.Bouchier 5, 1126 Sofia, Bulgaria

Abstract: The notion of Scott consequence system (briefly, S-system) was introduced by D. Vakarelov in [32] in an analogy to a similar notion given by D. Scott in [26]. In part one of the paper we study the category SSyst of all S-systems and all their morphisms. We show that the category DLat of all distributive lattices and all lattice homomorphisms is isomorphic to a reflective full subcategory of the category SSyst. Extending the representation theory of D. Vakarelov [32] for S-systems in P-systems, we develop an isomorphism theory for S-systems and for Tarski consequence systems. In part two of the paper we prove that the separation theorem for S-systems is equivalent in ZF to some other separation principles, including the separation theorem for filters and ideals in Boolean algebras and separation theorem for convex sets in convexity spaces.

DOI: 10.3233/FI-1998-33103

Journal: Fundamenta Informaticae, vol. 33, no. 1, pp. 43-70, 1998