Abstract: With extension processes we associate a partial structure and an extension process, a way of adding new elements to the already existing ones and/or a way for systematically modifying the already existing ones. In many cases the partial structure and the extension process define the final structure e.g. symmetries, incidences, types of subsets etc., but in other cases the result is ambiguous, there may be more than one non-isomorphic resulting structures. From an abstract point of view the question arises what kind of basic properties of partial structures, extension processes and extended structures can be identified. Some basic questions to be considered: does an extension process always define the final structure, or does the final structure depend on the partial structure as well? When can different extension processes lead to the same final structure? If a final structure is given, what kind of partial structures and extension processes can lead to it? Can we always find canonical partial structures? The lecture will provide examples and certain known partial results from geometry and biology. We will take a look at free geometries, look at homology and analogy in biological systems. |