Some properties of basic families of subsets

A basic system is a nonempty collection of finite incomparable subsets of a set such that for any two subsets or bases in the collection, any element of one basis can be replaced by some element of the other to give another basis in the collection. In a basic system, any subset of one basis can be bijectively exchanged for distinct elements of another; for a finite set, basis complements also have these properties; and certain conditions will guarantee that two such systems on the same set will contain a common basis. All proofs are new, elementary, and set-theoretic. In addition, they suggest efficient algorithmic procedures whose efficiencies are calculated.