On the structure of discrete systems

Abstract

The intention of this article is to present a general, uniform and concise mathematical framework for modelling of systems, especially discrete systems. As basic structures families and relations, defined as families of families represented in parameterized form, are used, which allows the representation of dual and polymorphic relations. On these structures structors are applied to obtain higher level structures, parts of these, and lower level structures from higher level structures. Considered are the π-product, selection of sub structures by properties, concatenations of relations subject to constraints. Treated are structures on index sets, topological structures, valuated structures, in particular fuzzy sets, sets of times as complete atomic boolean lattices with ordered atoms and induced orderings on the times, coarsenings of a time set, processes and their interactions, refinement and coarsenings of processes, and variables with their assignment operators. Most of our definitions are more general than those in literature. Relationships between various systems in applications are pointed out and illustrated by examples.