caa a22 4500 445344660 CHVBK 20180317142825.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s00233-010-9259-5 doi (NATIONALLICENCE)springer-10.1007/s00233-010-9259-5 An algorithm to compute ω -primality in a numerical monoid [Elektronische Daten] [David Anderson, Scott Chapman, Nathan Kaplan, Desmond Torkornoo] Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a 1⋅⋅⋅a t, where each a i is an atom, there is a T⊆{1,2, ,t} with |T|≤n such that x∣∏k∈T a k. The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1, ,2n−1〉, where n≥3, and 〈n,n+1, ,2n−2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing $\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}$. Springer Science+Business Media, LLC, 2010 Numerical Monoid nationallicence Non-unique factorization nationallicence Omega-primality nationallicence Anderson David Department of Mathematics, University of Tennessee, 37996-1300, Knoxville, TN, USA aut Chapman Scott Department of Mathematics and Statistics, Sam Houston State University, Box 2206, 77341-2206, Huntsville, TX, USA aut Kaplan Nathan Department of Mathematics, Princeton University, 08544-1000, Princeton, NJ, USA aut Torkornoo Desmond Department of Mathematics, University of Richmond, 28 Westhampton Way, 23173, Richmond, VA, USA aut Semigroup Forum Springer-Verlag 82/1(2011-02-01), 96-108 0037-1912 82:1<96 2011 82 233 https://doi.org/10.1007/s00233-010-9259-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00233-010-9259-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Anderson David Department of Mathematics, University of Tennessee, 37996-1300, Knoxville, TN, USA aut NATIONALLICENCE 700 1- Chapman Scott Department of Mathematics and Statistics, Sam Houston State University, Box 2206, 77341-2206, Huntsville, TX, USA aut NATIONALLICENCE 700 1- Kaplan Nathan Department of Mathematics, Princeton University, 08544-1000, Princeton, NJ, USA aut NATIONALLICENCE 700 1- Torkornoo Desmond Department of Mathematics, University of Richmond, 28 Westhampton Way, 23173, Richmond, VA, USA aut NATIONALLICENCE 773 0- Semigroup Forum Springer-Verlag 82/1(2011-02-01), 96-108 0037-1912 82:1<96 2011 82 233 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer