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1. CMB 2010 (vol 54 pp. 39)

Chapman, S. T.; García-Sánchez, P. A.; Llena, D.; Marshall, J.
 Elements in a Numerical Semigroup with Factorizations of the Same Length Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions. Keywords:numerical monoid, numerical semigroup, non-unique factorizationCategories:20M14, 20D60, 11B75

2. CMB 2009 (vol 52 pp. 598)

Moreno, M. A.; Nicola, J.; Pardo, E.; Thomas, H.
 Numerical Semigroups That Are Not Intersections of $d$-Squashed Semigroups We say that a numerical semigroup is \emph{$d$-squashed} if it can be written in the form $$S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$ for $N,a_1,\dots,a_d$ positive integers with $\gcd(a_1,\dots, a_d)=1$. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular. Recent works by Rosales \emph{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. Keywords:numerical semigroup, squashed semigroup, proportionally modular semigroupCategories:20M14, 06F05, 46L80

3. CMB 2008 (vol 51 pp. 134)

Rosales, J. C.; Garc\'{\i}a-Sánchez, P. A.
 Numerical Semigroups Having a Toms Decomposition We show that the class of system proportionally modular numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition. Categories:20M14, 11D75
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