Constructible Sets and Algebras

Consider an initial family of subsets of some universe X. We define a 1-constructible family to be the family obtained from the initial family by taking all pairwise unions, pairwise intersections, and complements. We then iterate the operation to obtain a k-constructible family for each integer k. Under some conditions, there exists a k such that all families beyond the k-constructible family are identical. In this case, the final family has an algebraic structure. When the universe X is finite, we characterize initial families that generate the power set of X. In addition, given any universe and any initial family, we describe the elements of any finite algebra constructible from the initial family. We use analysis to prove that any finite algebra, A, has a unique maximum partition that generates A. We calculate the size of a smallest generating family based on the order of the algebra and provide a construction for a generating family of that size. We prove that there are multiple generating families of the smallest size and count the number of such generating families for algebras of size up to 2⁵.