On the primality and elasticity of algebraic valuations of cyclic free semirings

A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number α, the additive monoid Mα of the evaluation semiring N0[α] is atomic. The atomic structure of both the additive and the multiplicative monoids of N0[α] has been the subject of several recent papers. Here we focus on the monoids Mα, and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when α is less than 1, the atoms of N0[α] are as far from being prime as they can possibly be. Then we establish some results about the elasticity of N0[α], including that when α is rational, the elasticity of Mα is full (this was previously conjectured by S. T. Chapman, F. Gotti, and M. Gotti).

[1]  Abraham Zaks.  Half-factorial-domains , 1980 .

[2]  Sophie Zhu,et al.  Factorizations in evaluation monoids of Laurent semirings , 2021, Communications in Algebra.

[3]  N. Baeth,et al.  Factorizations in upper triangular matrices over information semialgebras , 2020, 2002.09828.

[4]  F. Gotti,et al.  Atomicity and boundedness of monotone Puiseux monoids , 2016, 1608.04044.

[5]  David F. Anderson,et al.  On the Elasticities of Krull Domains with Finite Cyclic Divisor Class Group , 2000 .

[6]  Marly Gotti,et al.  Factorizations in reciprocal Puiseux monoids , 2021, 2112.04048.

[7]  S. Chapman,et al.  Full Elasticity in Atomic Monoids and Integral Domains , 2006 .

[9]  F. Halter-Koch.  Finiteness theorems for factorizations , 1992 .

[10]  A. Geroldinger,et al.  Long Sets of Lengths With Maximal Elasticity , 2017, Canadian Journal of Mathematics.

[11]  Jyrko Correa-Morris,et al.  On the additive structure of algebraic valuations of cyclic free semirings , 2020 .

[12]  F. Gotti,et al.  The elasticity of Puiseux monoids , 2017, Journal of Commutative Algebra.

[13]  Scott T. Chapman,et al.  Bi-atomic classes of positive semirings , 2021, Semigroup Forum.

[14]  D. Grynkiewicz.  The Characterization of Finite Elasticities , 2020, 2012.12757.

[15]  Wolfgang Hassler,et al.  Local tameness of v-noetherian monoids , 2008 .

[16]  M. Fontana,et al.  Elastic properties of some semirings defined by positive systems , 2009 .

[17]  Harold Polo,et al.  On the atomic structure of exponential Puiseux monoids and semirings , 2020, Communications in Algebra.

[18]  PETE L. CLARK,et al.  Factorization in Integral Domains I , 2009 .

[19]  Felix Gotti,et al.  On semigroup algebras with rational exponents , 2021, Communications in Algebra.

[20]  D. R. Curtiss.  Recent Extentions of Descartes' Rule of Signs , 1918 .

[21]  David F. Anderson,et al.  Bounded and finite factorization domains , 2020, 2010.02722.

[22]  S. Chapman,et al.  Factorization invariants of Puiseux monoids generated by geometric sequences , 2019, Communications in Algebra.

[23]  J. Steffan.  Longueurs des décompositions en produits d'éléments irréductibles dans un anneau de Dedekind , 1986 .

[24]  David F. Anderson,et al.  Elasticity of factorizations in integral domains , 1992 .

[25]  Alexander J. Barrios.  Elasticity of Factorization in Number Fields , 2012 .

[26]  F. Gotti.  Increasing positive monoids of ordered fields are FF-monoids , 2016, Journal of Algebra.

[27]  F. Gotti.  The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid , 2020, Journal of Algebra and Its Applications.