1. Introduction. Various types of multiple-valued dynamical systems have appeared in the literature over the past twenty years (2). The motivations for such generalizations are varied, but among others, control theory plays a significant role. In a recent paper (2), Bushaw has introduced an abstract approach which consolidates a broad class of these dynamical systems. The model is a quasi-ordered set (E, ρ); i.e. Ε is a non-empty set and ρ is a reflexive and transitive relation on E. E represents the ‘event’ space of some real-world system and ρ represents some class of ‘admissible’ inputs on the system; i.e. (e1, e2) ∈ ρ if and only if there is an admissible input function which ‘shifts’ the system from event e1 to event e2. ρ is called the attainability relation on E and the statement ‘ an event e2 is attainable from an event e1 ’ means that (e1, e2) ∈ ρ. Adopting the terminology of (2), we shall refer to the quasi-ordered set (E, ρ) as a polysystem.