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Numerical Semigroups That Are Not Intersections of d-Squashed Semigroups
Published online by Cambridge University Press: 20 November 2018
Abstract
We say that a numerical semigroup is $d$-squashed if it can be written in the form
for $N$, ${{a}_{1}}\,,\,.\,.\,.\,,\,{{a}_{d}}$ positive integers with $\gcd \left( {{a}_{1}},\,.\,.\,.\,,{{a}_{d}} \right)\,=\,1$. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular.
Recent works by Rosales et al. give a concrete example of a numerical semigroup that cannot be written as an intersection of 2-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of 2-squashed semigroups. We also will prove the same result for 3-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of $d$-squashed semigroups for any fixed $d$, and we prove some partial results towards this conjecture.
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- Copyright © Canadian Mathematical Society 2009
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