Media theory

Abstract

This paper reviews and extends previous results of the first author concerning a particular semigroup of transformations on a finite set of states. The noun ‘medium’ given to this semigroup stems from an important application in which the transformations formalize the effects, on an individual, of ‘tokens’ of information delivered by the environment—i.e. the ‘medium’—thereby modifying his or her opinions. The axioms containing the semigroup actually capture a wide variety of examples ranging from convex analysis to combinatorics. A common characteristic is that any transformation of a state—if it is effective—leaves a trace which is a partially defining feature of the state produced. For instance, the family of all strict partial orders on a finite set, equipped with the set of transformations consisting in adding (or removing) an ordered pair to (or from) a partial order to form another partial order is an instance of a medium. As suggested by this example, while these transformations are never one-to-one functions, each transformation has a unique ‘reverse’ transformation undoing its actions. We introduce the concepts of ‘orientation’ and ‘closure’ for a medium and derive some consequences. A recently published application of media theory to the analysis of opinion polls data is briefly discussed.