The object of this paper is to introduce some types of formal objects which may serve the dual purpose of representing natural dialogues as well as being models of a mathematical theory of dialogues. When trying to develop a mathematical theory one way to proceed is by characterizing its set-theoretical models and this can be initiated by characterizing their basic structure. Lacking a definite proposal as to the form the theory should have the best strategy seems to be to define and compare objects with different (basic) structures which may serve as models of the theory. This is the strategy I use. To put it roughly: What I attempt to do here is to reduce the notion of dialogue, not to define it.¹