Media Theory – Examples and Representations

©
Sergei Ovchinnikov

Mathematics Department

San Francisco State University

San Francisco, CA 94132

Abstract

J.-Cl. Falmagne introduced the concept of a medium in 1997 as a special case of a general token system. The initial motivation for his work came from some problems in social and behavioral sciences. For instance, in the framework of preference evolution, individuals may be asked to provide their preference relations (called states) over a given set of alternatives. As time evolves, the individuals are bombarded with a discrete stream of unobservable ‘particles’ of information (called tokens) which may alter the individual states. The combinatorial part of media theory deals with a particular semigroup of transformations of an abstract set of states V. This semigroup is generated by the set of tokens. Each token is a transformation of V, which is not a bijection but has a unique reverse. It is also postulated that this semigroup is transitive on V. There are two other axioms that ensure the consistency of the transformations producing a given state.

In this talk we will present an approach to media theory without making
an assumption that the sets of states and tokens are finite. We will show
that any medium is isomorphic to a medium of well graded family of sets and
establish the uniqueness of such a representation. A medium can be represented
in a natural way by its graph. In the talk we will characterize graphs that
represent media. Our presentation will have a distinctive geometric flavor.