2023/04/30 by Khalid Ajran, Ajran, Khalid, Felix Gotti +1
Computer Science · Mathematics · #05C90 #20M13 #Advanced Algebra and Logic #Combinatorics (math.CO) #FOS: Mathematics #Primary: 11Y05 #Rings, Modules, and Algebras #Secondary: 06B25 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2305.00413
openalex publication_date 2023/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a join semilattice S with a minimum 0, the quarks (also called atoms in order theory) are the elements that cover 0, and for each x ∈ S ∖ \0\ a factorization (into quarks) of x is a minimal set of quarks whose join is x. If every element x ∈ S ∖ \0\ has a factorization, then S is called factorizable. If for each x ∈ S ∖ \0\, any two factorizations of x have equal (resp., distinct) size, then we say that S is half-factorial (resp., length-factorial). Let B_ℕ be the Boolean lattice consisting of all finite subsets of ℕ under intersections and unions. Here we study factorizations into quarks in join subsemilattices of B_ℕ, focused on the notions of half-factoriality and length-factoriality. We also consider the unique factorization property, which is the most special and relevant type of half-factoriality, and the elasticity, which is an arithmetic statistic that measures the deviation from half-factoriality.