Re-visiting axioms of information systems

Based on the investigation of the axioms used in the theory of information system, the notion of simplified continuous information system is introduced which is considered as a generalization of continuous information systems. Just like the continuous information system, it is also a concrete representation of continuous domains and categorically equivalent to the category of continuous domains and Scott continuous functions. What is more, all types of information systems studied in the literature so far turn out to be special cases of the new type. By defining linear information systems and their entailment functions, we show that the states of a Scott-style information system are exactly the fixpoints of its entailment function. Some axioms about an entailment relation can be characterized by the properties of the corresponding entailment function, and in particular, the simplified continuous information systems are solely determined by the idempotence of entailment functions.