2020/05/04 by Salvatore Tringali, Tringali, Salvatore
Chemistry · Computer Science · Mathematics · #16U30 #20M13. Secondary 13A05 #20M14 #Alphabet #Chemistry #Combinatorics #Commutative Algebra (math.AC) #Commutative property #Discrete mathematics #FOS: Mathematics #Factorization #Finite Group Theory Research #Free monoid #Mathematics #Monoid #Primary 20M10 #Product (mathematics) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #Stereochemistry #Unit (ring theory) #math.AC #math.RA #msc:13A05 #msc:16U30 #msc:20M10 #msc:20M13. #msc:20M14 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2005.01681
24 pp.; no figures. Comments are very welcome
arxiv created 2020/05/04 · openalex publication_date 2020/05/04 · arxiv updated 2020/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H^× be the group of units of a multiplicatively written monoid H. We say H is acyclic if xyz ≠ y for all x, y, z ∈ H with x ∉ H^× or z ∉ H^×; unit-cancellative if yx ≠ x ≠ xy for all x, y ∈ H with y ∉ H^×; f.g.u. if there is a finite set A ⊆ H such that every non-unit of H is a finite product of elements of the form uav with u, v ∈ H^× and a ∈ A; l.f.g.u. if, for each x ∈ H, the smallest divisor-closed submonoid of H containing x is f.g.u; and atomic if every non-unit can be written as a finite product of atoms, where an atom is a non-unit that does not factor into a product of two non-units. We generalize to l.f.g.u. or acyclic l.f.g.u. monoids a few results so far only known for unit-cancellative l.f.g.u. commutative monoids (cancellative monoids are unit-cancellative, and a commutative monoid is unit-cancellative if and only if it is acyclic). In particular, we prove the following: \bullet If H is an atomic l.f.g.u. monoid, then every non-unit has only finitely many factorizations (into atoms) that are "minimal" and "pairwise non-equivalent" (with respect to some naturally defined relations on the free monoid over the "alphabet" of atoms). \bullet If H is an acyclic l.f.g.u. monoid, then it is atomic; and moreover, each element has only finitely many "pairwise non-equivalent" factorizations if we additionally assume H to be commutative.