Filters and mathematical systems

A filter in a set is any device which passes or does not pass each element in a set. The action of a filter is the dichotomy (A,B) of the base set where A is the set of elements passed or accepted and B is the complement of A. This innocent appearing definition which I first stated in 1967 is both general and suggestive of many applications. In particular, I chose the definition by direct abstraction from the filters of chemistry, electronics, cigarettes, and air conditioning. If I should not have differentiated between the filter as a device and its action, then the concept would be reduced to the algebra of sets. In the next section I will give a number of examples of filters and a short summary of a rudimentary theory. Thereafter I proceed to discussions of an assortment of generalizations I have made in which filters as such will appear mainly to indicate that they are ubiquitous. I will however, concentrate on the concepts which I know to be of importance in general systems theories. My approach has not followed the lines of making mathematical models of systems but to study mathematics as a system. While I have many years of work staked on the utility of such an approach, I do not pretend to have demonstrated to the satisfaction of many that applications are indeed possible. The semantics of mathematics, while more circumscribed than that of languages is not to be trivially described even if we choose to ignore its dynamic aspects. Even such a seemingly simple distinction as the one between abstraction and generalization have been confused partially because of the reluctance to admit that a generalization embraces the specific concept generalized. For example all groups are semigroups (by a poor choice of terminology) and thus semigroups are a generalization of groups, not an abstraction. On the other hand semigroups may be defined by abstracting certain conditions required of groups and in this sense they are the result of an abstraction process.