Minimal Constructible Sets
Jorge Garcia, Rosemarie Bongers, Jonathan Detgen, Walter Morales

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 (-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 -minimal constructible families; one major result is that every finite algebra of sets has a generating family which is -minimal constructible for all…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Algebra and Logic · Computability, Logic, AI Algorithms
