Vol. 312, No. 2, 2021

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On the arithmetic of power monoids and sumsets in cyclic groups

Austin A. Antoniou and Salvatore Tringali

Vol. 312 (2021), No. 2, 279–308
Abstract

Let H be a multiplicatively written monoid with identity 1H (in particular, a group), and denote by 𝒫fin,×(H) the monoid obtained by endowing the collection of all finite subsets of H containing a unit with the operation of setwise multiplication (X,Y ){xy : x X,y Y }. We study fundamental features of the arithmetic of this and related structures, with a focus on the submonoid, 𝒫fin,1(H), of 𝒫fin,×(H) consisting of all finite subsets of H containing the identity.

Among others, we establish that 𝒫fin,1(H) is atomic (i.e., each nonunit is a product of atoms) if and only if 1Hx2x for every x H {1H}. Then we prove that 𝒫fin,1(H) is BF (i.e., it is atomic and every element has factorizations of bounded length) if and only if H is torsion-free; and we show how to transfer these conclusions from 𝒫fin,1(H) to 𝒫fin,×(H) through the machinery of equimorphisms.

Next, we introduce a suitable notion of “minimal factorization” (and investigate its behavior with respect to equimorphisms) to account for the fact that monoids may have nontrivial idempotents, in which case standard definitions from factorization theory degenerate. Accordingly, we obtain necessary and sufficient conditions for 𝒫fin,×(H) to be BmF (meaning that each nonunit has at least one minimal factorization and all such factorizations are bounded in length); and for 𝒫fin,1(H) to be BmF, HmF (i.e., a BmF-monoid where all the minimal factorizations of a given element have the same length), or minimally factorial (i.e., a BmF-monoid where each nonunit element has an essentially unique minimal factorization). Finally, we prove how to realize certain intervals as sets of minimal lengths in 𝒫fin,1(H).

Many proofs come down to considering sumset decompositions in cyclic groups, so giving rise to an intriguing interplay with arithmetic combinatorics.

Keywords
BF-monoids, decompositions into atoms, irreducibles, minimal factorizations, nonunique factorization, power monoids, product sets, sumsets
Mathematical Subject Classification 2010
Primary: 11B30, 11P70, 20M13
Secondary: 11B13
Milestones
Received: 13 August 2019
Revised: 3 October 2020
Accepted: 30 November 2020
Published: 31 August 2021
Authors
Austin A. Antoniou
Department of Mathematics
The Ohio State University
Columbus, OH 43202
United States
Salvatore Tringali
Institute for Mathematics and Scientific Computing
University of Graz
8010 Graz
Austria
School of Mathematical Sciences
Hebei Normal University
Hebei 050024
China