Rewriting systems over similarity and generalized pseudometric spaces and their properties

We present a study of confluence and related properties of fuzzy relations defined over similarity spaces. The ordinary confluence is an essential property of relations connected to the idea of rewriting and substituting which appears in abstract rewriting systems. This paper is a continuation of our previous paper, where we have introduced analogous notions related to substitutability in graded setting using residuated lattices as structures of truth degrees, leaving the ordinary notions a particular case when the underlying structure is the two-valued Boolean algebra. In this paper, we further extend our previous results by developing the notions of confluence and related properties respecting a given similarity relation. We also present definitions of confluence and related properties of relations on a generalized pseudometric spaces. Using the well-known link between generalized pseudometrics and similarities, we describe the connection of the notions defined on generalized pseudometric spaces to the corresponding notions on similarity spaces. The introduced notions are further illustrated by example of rewriting based on graded if–then rules.