On strongly primary monoids, with a focus on Puiseux monoids

Abstract

Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. It is well-known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is Noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux monoids are primary monoids, and we provide conditions ensuring that they are strongly primary. Then we study local and global tameness of strongly primary Puiseux monoids; most notably, we establish an algebraic characterization of when a Puiseux monoid is globally tame. Moreover, we obtain a result on the structure of sets of lengths of all locally tame strongly primary monoids.

25 pages. It will appear in Journal of Algebra

References in corpus (12)

• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
• Increasing positive monoids of ordered fields are FF-monoids
• Atomicity and boundedness of monotone Puiseux monoids
• The Realization Problem for Delta Sets of Numerical Semigroups
• Structural properties of subadditive families with applications to factorization theory
• On the Atomic Structure of Puiseux Monoids
• Long sets of lengths with maximal elasticity
• A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems
• Arithmetical invariants of local quaternion orders
• On the system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid
• Realizable sets of catenary degrees of numerical monoids
• On the arithmetic of Mori monoids and domains

Cited by in corpus (12)

• Factorizations in upper triangular matrices over information semialgebras
• On the additive structure of algebraic valuations of polynomial semirings
• Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid
• Semigroup rings as weakly Krull domains
• On the atomic structure of exponential Puiseux monoids and semirings
• A conjecture by Bienvenu and Geroldinger on power monoids
• On the atomic structure of torsion-free monoids
• Length-Factoriality and Pure Irreducibility
• Semigroups of ideals and isomorphism problems
• Factorization Theory in Commutative Monoids
• A characterization of finite factorization positive monoids
• Atomicity and Factorization of Puiseux Monoids