Equal catenary degree of an element
Definition
Let \(S\) be a numerical semigroup, \(s \in S\) and \(\mathbf{Z}(s)\) its set of factorizations. It is defined the equal catenary degree of \(s\) in \(S\), denoted by \(EqC(S)\), as the least \(N\) such that for any two factorizations \(x,y \in \mathbf{Z}(s)\) with the same length, there exist an N-chain joining them with all the factorizations of the same length. Equivalently, if \(l_1, \ldots, l_n\) are the lengths of the factorizations and \(\mathbf{Z}_{l_i}\) are the factorizations of length \(l_i\) for \(i \in \{1, \ldots, n\}\), then the equal catenary degree of \(s\) is the maximum of the catenary degrees of \(\mathbf{Z}_{l_1}, \ldots, \mathbf{Z}_{l_n}\).
Examples
\(\circ\) Let \(S = \langle 7, 12, 15 \rangle\) and \(s = 100\). If \((\alpha, \beta, \gamma) \in \mathbf{Z}(100)\), then
\[ 7 \cdot \alpha + 12 \cdot \beta + 15 \cdot \gamma = 100, \]
where necessarily \(0 \le \alpha \le 7, ~ 0 \le \beta \le 8, ~ 0 \le \gamma \le 6\),
deducing that
\[ \mathbf{Z}(100) = \{(4, 6, 0), (7, 3, 1), (10, 0, 2), (1, 4, 3), (4, 1, 4) \}. \]
Their lengths are \((l_1, \ldots, l_5) = (8,9,10,11,12)\) and
\[ \mathbf{Z}_8 = \{(1,4,3)\}, ~~ \mathbf{Z}_9 = \{(4,1,4)\}, ~~ \mathbf{Z}_{10} = \{(4,6,0)\}, ~~ \mathbf{Z}_{11} = \{(7,3,1)\}, ~~ \mathbf{Z}_{12} = \{(10, 0, 2)\}. \]
Since each set has one factorization, \(EqC(100) = 0\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 20, 39, 96, 105, 126, 130 \rangle\), in GAP:
gap> S := NumericalSemigroup(20, 39, 96, 105, 126, 130);
<Numerical semigroup with 6 generators>Given a set of factorizations \(Is\), the function EqualCatenaryDegreeOfSetOfFactorizations returns the equal catenary degree of \(Is\).
gap> 600 in S;
true
gap> Is := Factorizations(600, S);
[ [ 30, 0, 0, 0, 0, 0 ], [ 0, 8, 3, 0, 0, 0 ], [ 3, 4, 4, 0, 0, 0 ],
[ 6, 0, 5, 0, 0, 0 ], [ 15, 5, 0, 1, 0, 0 ], [ 18, 1, 1, 1, 0, 0 ],
[ 0, 10, 0, 2, 0, 0 ], [ 3, 6, 1, 2, 0, 0 ], [ 6, 2, 2, 2, 0, 0 ],
[ 9, 0, 0, 4, 0, 0 ], [ 12, 6, 0, 0, 1, 0 ], [ 15, 2, 1, 0, 1, 0 ],
[ 0, 7, 1, 1, 1, 0 ], [ 3, 3, 2, 1, 1, 0 ], [ 6, 1, 0, 3, 1, 0 ],
[ 0, 4, 2, 0, 2, 0 ], [ 3, 0, 3, 0, 2, 0 ], [ 3, 2, 0, 2, 2, 0 ],
[ 0, 3, 0, 1, 3, 0 ], [ 0, 0, 1, 0, 4, 0 ], [ 4, 10, 0, 0, 0, 1 ],
[ 7, 6, 1, 0, 0, 1 ], [ 10, 2, 2, 0, 0, 1 ], [ 13, 0, 0, 2, 0, 1 ],
[ 1, 1, 1, 3, 0, 1 ], [ 10, 1, 0, 1, 1, 1 ], [ 7, 2, 0, 0, 2, 1 ],
[ 17, 0, 0, 0, 0, 2 ], [ 2, 5, 0, 1, 0, 2 ], [ 5, 1, 1, 1, 0, 2 ],
[ 2, 2, 1, 0, 1, 2 ], [ 0, 0, 0, 2, 0, 3 ], [ 4, 0, 0, 0, 0, 4 ] ]
gap> EqualCatenaryDegreeOfSetOfFactorizations(Is);
5References
https://gap-packages.github.io/ numericalsgps.