Mediatic Graphs

The core concept of this paper can occur in the guise of various representations. Four of them are relevant here, the last one being new:

• 1. A MEDIUM, that is, a semigroup of transformations on a set of states, constrained by strong axioms (see Eppstein, Falmagne, & Ovchinnikov, 2008; Falmagne, 1997; Falmagne & Ovchinnikov, 2002).

• 2. An ISOMETRIC SUBGRAPH OF THE HYPERCUBE, OR “PARTIAL CUBE.” By “isometric”, we mean that the distance between any two vertices of the subgraph is identical to the distance between the same two vertices in the hypercube (Djoković, 1973; Graham & Pollak, 1971). Each state of the medium is mapped to a vertex of the graph, and each transformation corresponds to an equivalence class of its arcs. Note that, as will become clear later on, no assumption of finiteness is made in this or in any of the other representation.

• 3. An ISOMETRIC SUBGRAPH OF THE INTEGER LATTICE. This representation is not exactly interchangeable with the preceding one. While it is true that any isometric subgraph of the hypercube is representable as an isometric subgraph of the integer lattice and vice versa, the latter representation lands in a space equipped with a considerable amount of structure. Notions of “lines”, “hyperplanes”, or “parallelism” can be legitimately defined if one wishes. Moreover, the dimension of the lattice representation is typically much smaller than that of the partial cube representing the same medium and so can be valuable in the representation of large media (see, in particular, Eppstein, 2005, in which an algorithm is described for finding the minimum dimension of a lattice representation of a partial cube).

• 4. A MEDIATIC GRAPH. Axiomatic definitions are usually regarded as preferable whenever feasible, and that is what is given here.