Minimal Constructible Sets

TL;DR

This paper investigates the recursive process of constructing set families through unions, intersections, and complements, characterizing the minimal steps and properties of the resulting families, with implications for algebra generation and connections to Baire's Theorem.

Contribution

It introduces the concept of $n$-minimal constructible families, characterizes the last family in the construction process, and proves that every finite algebra has an $n$-minimal generating family for all natural $n$.

Findings

Every finite algebra of sets has an $n$-minimal constructible generating family.

The minimum number of steps to generate an algebra is computed.

A connection between the construction process and Baire's Theorem is established.

Abstract

Given an initial family of sets, we may take unions, intersections and complements of the sets contained in this family in order to form a new collection of sets; our construction process is done recursively until we obtain the last family. Problems encountered in this research include the minimum number of steps required to arrive to the last family as well as a characterization of that last family; we solve all those problems. We also define a class of simple families ($n$-minimal constructible) and we analyze the relationships between partitions and separability (our new concept) that lead to interesting results such as finding families based on partitions that generate finite algebras. We prove a number of new results about $n$-minimal constructible families; one major result is that every finite algebra of sets has a generating family which is $n$-minimal constructible for all natural $n$ and we compute the minimum number of steps required to generate an algebra. Another interesting result is a connection between this construction process and Baire's Theorem. This work has a number of possible applications, particularly in the fields of economics and computer science.