Adjacent catenary degree of an element

Definition

Let \(S\) be a numerical semigroup, \(s \in S\) and \(\mathbf{Z}(s)\) its set of factorizations. Let us consider \(l_1, \ldots, l_t\) the lengths of the factorizations of \(s\) in \(S\) and \(\mathbf{Z}_{l_i}\) the set of factorizations of \(s\) with length \(l_i\). It is defined the adjacent catenary degree of \(s\) in \(S\), denoted by \(AdjC(s)\), as

\[ AdjC(s) = \max \{d(\mathbf{Z}_{l_i}, \mathbf{Z}_{l_{i+1}}) ~ | ~ i \in \{1, \ldots, n-1\} \}, \]

where \(d(\mathbf{Z}_{l_i}, \mathbf{Z}_{l_{i+1}})\) is defined as

\[ d(\mathbf{Z}_{l_i}, \mathbf{Z}_{l_{i+1}}) = \min \{d(x,y) ~ | ~ x \in \mathbf{Z}_{l_i}, ~ y \in \mathbf{Z}_{l_{i+1}}\}, \]

with \(d(x,y)\) denoting the distance between \(x\) and \(y\).

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

\[ \mathbf{Z}_8 = \{(1,4,3)\}, ~~ \mathbf{Z}_9 = \{(4,1,4)\}, ~~ \mathbf{Z}_{10} = \{(4,6,0)\}, ~~ \mathbf{Z}_{11} = \{(7,3,1)\}, ~~ \mathbf{Z}_{12} = \{(10, 0, 2)\}. \]

Finally,

\[ d(\mathbf{Z}_8, \mathbf{Z}_9) = d((1,4,3), (4,1,4)) = 4, ~~ d(\mathbf{Z}_9, \mathbf{Z}_{10}) = d((4,1,4), (4,6,0)) = 5 \]

\[ d(\mathbf{Z}_{10}, \mathbf{Z}_{11}) = d((4,6,0), (7,3,1)) = 4, ~~ d(\mathbf{Z}_{11}, \mathbf{Z}_{12}) = d((7,3,1), (10, 0, 2)) = 4, \]

concluding that \(AdjC(100) = 5\).

Examples with GAP

The following example is made with the package NumericalSgps in GAP.

\(\diamond\) Let \(S = \langle 13, 14, 33, 64, 76 \rangle\), in GAP:

gap> S := NumericalSemigroup(13, 14, 33, 64, 76);
<Numerical semigroup with 5 generators>

Given a set of factorizations \(Is\), the function AdjacentCatenaryDegreeOfSetOfFactorizations returns the adjacent catenary degree of \(Is\). Given a numerical semigroup \(S\) and an element \(n \in S\), the function Factorizations returns the set of factorizations of \(n\) in \(S\).

gap> 300 in S;
true
gap> Is := Factorizations(S, 300);
[ [ 22, 1, 0, 0, 0 ], [ 8, 14, 0, 0, 0 ], [ 13, 7, 1, 0, 0 ], [ 18, 0, 2, 0, 0 ],
  [ 4, 13, 2, 0, 0 ], [ 9, 6, 3, 0, 0 ], [ 0, 12, 4, 0, 0 ], [ 5, 5, 5, 0, 0 ],
  [ 1, 4, 7, 0, 0 ], [ 16, 2, 0, 1, 0 ], [ 2, 15, 0, 1, 0 ], [ 7, 8, 1, 1, 0 ],
  [ 12, 1, 2, 1, 0 ], [ 3, 7, 3, 1, 0 ], [ 8, 0, 4, 1, 0 ], [ 10, 3, 0, 2, 0 ],
  [ 1, 9, 1, 2, 0 ], [ 6, 2, 2, 2, 0 ], [ 2, 1, 4, 2, 0 ], [ 4, 4, 0, 3, 0 ],
  [ 0, 3, 2, 3, 0 ], [ 14, 3, 0, 0, 1 ], [ 0, 16, 0, 0, 1 ], [ 5, 9, 1, 0, 1 ],
  [ 10, 2, 2, 0, 1 ], [ 1, 8, 3, 0, 1 ], [ 6, 1, 4, 0, 1 ], [ 2, 0, 6, 0, 1 ],
  [ 8, 4, 0, 1, 1 ], [ 4, 3, 2, 1, 1 ], [ 0, 2, 4, 1, 1 ], [ 2, 5, 0, 2, 1 ],
  [ 6, 5, 0, 0, 2 ], [ 2, 4, 2, 0, 2 ], [ 0, 6, 0, 1, 2 ], [ 3, 0, 1, 0, 3 ] ]
gap> AdjacentCatenaryDegreeOfSetOfFactorizations(Is);
14

References