Structures with features

What do we look at first when checking a proof or a student's exercise? What is important in recognizing someone else's face? What are the important things that make us realize we are looking at a (mathematical) group? How do we look at a building? How do pilots navigate safely with so many instruments to look at? All of these are examples involving structures with features. The concept seems to arise in so many different contexts but not to have been formalized. This is true despite the use of Feature Set Theory by cognitive psychologists (see [18] and related work by Christopher Alexander (see [2] and [3]) which both date back to the mid-seventies. We use the examples to suggest how features are important in the process of abstraction from known structures, and how and when formal descriptions correspond to the world. We consider the operations of modifying, adding and deleting features in a structure and the idea of isomorphisms and other similarity relations between structures with features. In particular we analyse what it means for classes of structures with features to be equivalent.