On the atomic structure of exponential Puiseux monoids and semirings

Abstract

We say that a Puiseux monoid is exponential provided that it is generated by some of the powers of a rational number. Here we study the atomic properties of exponential Puiseux monoids and semirings. First, we characterize atomic exponential Puiseux monoids, and we prove that the finite factorization property, the bounded factorization property, and the ACCP coincide in this context. Then we proceed to offer necessary and sufficient conditions for an exponential Puiseux monoid to satisfy the ACCP. We conclude by describing the exponential Puiseux monoids that are semirings.

Keywords:

• ACCP
• atomic monoids
• atomicity
• Puiseux monoids
• Puiseux semirings
• rational cyclic semirings

2010 MATHEMATICS SUBJECT CLASSIFICATION Primary:

• 20M13
• Secondary: 16Y60, 06F05, 20M14

Acknowledgment

The authors want to thank Felix Gotti for his mentorship and guidance during the preparation of this paper, and anonymous referees for their careful revision. While working on this manuscript, the third author was supported by the University of Florida Mathematics Department Fellowship.