Projection of a medium

Learning spaces, partial cubes, and preference orderings are just a few of the many structures that can be captured by a ‘medium,’ a set of transformations on a possibly infinite set of states, constrained by four strong axioms. In this paper, we introduce a method for summarizing an arbitrary medium by gathering its states into equivalence classes and treating each equivalence class as a state in a new structure. When the new structure is also a medium, it can be characterized as a projection of the original medium. We show that any subset of tokens from an arbitrary medium generates a projection, and that each state in the projection determines a submedium.