The predicate completion of a partial information system

Abstract

AbstractOriginally,partial information systemswere introduced as a means of providing a representation of the Smyth powerdomain in terms of order convex substructures of an information-based structure. For every partial information system 𝕊, there is a new partial information system that is natrually induced by the principal lowersets of the consistency predicate for 𝕊. In this paper, we show that this new system serves as acompletionof the parent system 𝕊 in two ways. First, we demonstrate that the induced system relates to the parent system 𝕊 in much the same way as the ideal completion of the consistency predicate for 𝕊 relates to the consistency predicate itself. Second, we explore the relationship between this induced system and the notion ofD-completions for posets. In particular, we show that this induced system has a “semi-universal” property in the category of partial information systems coupled with the preorder analog of Scott-continuous maps that is induced by the universal property of theD-completion of the principal lowersets of the consistency predicate for the parent system 𝕊.