Monotone catenary degree of an element
Definition
Let \(S\) be a numerical semigroup, \(s \in S\) and \(\mathbf{Z}(s)\) its set of factorizations. Is defined the monotone catenary degree of \(s\) in \(S\), denoted by \(MonC(s)\), as the least \(N\) such that for any two factorizations \(x,y \in \mathbf{Z}(s)\) with lengths \(|x| \le |y|\), there exists an N-chain of factorizations with non-decreasing lengths joining them.
It can be proven that the monotone catenary degree of is the maximum of the adjacent catenary degree and the equal catenary degree of \(s\) in \(S\).
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, \(EqC(S) = 0\). On the other hand,
\[ d(\mathbf{Z}_8, \mathbf{Z}_9) = d((1,4,3), (4,1,4)) = 4, ~~ d(\mathbf{Z}_9, \mathbf{Z}_{10}) = d((4,1,4), (4,6,0)) = 5 \]
\[ d(\mathbf{Z}_{10}, \mathbf{Z}_{11}) = d((4,6,0), (7,3,1)) = 4, ~~ d(\mathbf{Z}_{11}, \mathbf{Z}_{12}) = d((7,3,1), (10, 0, 2)) = 4, \]
concluding that \(AdjC(100) = 5\) and \(MonC(S) = 5\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 39, 43, 46, 50, 53, 58, 79, 102, 126 \rangle\), in GAP:
gap> S := NumericalSemigroup(39, 43, 46, 50, 53, 58, 79, 102, 126);
<Numerical semigroup with 9 generators>Given a set of factorizations \(Is\), the function MonotoneCatenaryDegreeOfSetOfFactorizations computes the monotone catenary degree of \(Is\). On the other hand, the functions AdjacentCatenaryDegreeOfSetOfFactorizations and EqualCatenaryDegreeOfSetOfFactorizations compute the adjacent catenary degree and the equal catenary degree of \(Is\), respectively.
gap> 361 in S;
true
gap> Is := Factorizations(361, S);
[ [ 1, 0, 7, 0, 0, 0, 0, 0, 0 ], [ 0, 5, 1, 2, 0, 0, 0, 0, 0 ],
[ 1, 4, 0, 3, 0, 0, 0, 0, 0 ], [ 2, 0, 5, 0, 1, 0, 0, 0, 0 ],
[ 0, 6, 0, 1, 1, 0, 0, 0, 0 ], [ 3, 0, 3, 0, 2, 0, 0, 0, 0 ],
[ 4, 0, 1, 0, 3, 0, 0, 0, 0 ], [ 0, 0, 1, 1, 5, 0, 0, 0, 0 ],
[ 0, 1, 0, 0, 6, 0, 0, 0, 0 ], [ 1, 4, 2, 0, 0, 1, 0, 0, 0 ],
[ 2, 3, 1, 1, 0, 1, 0, 0, 0 ], [ 3, 2, 0, 2, 0, 1, 0, 0, 0 ],
[ 2, 4, 0, 0, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 5, 1, 1, 0, 0, 0 ],
[ 4, 1, 1, 0, 0, 2, 0, 0, 0 ], [ 5, 0, 0, 1, 0, 2, 0, 0, 0 ],
[ 0, 0, 2, 2, 1, 2, 0, 0, 0 ], [ 0, 1, 1, 1, 2, 2, 0, 0, 0 ],
[ 1, 0, 0, 2, 2, 2, 0, 0, 0 ], [ 0, 2, 0, 0, 3, 2, 0, 0, 0 ],
[ 0, 3, 0, 0, 0, 4, 0, 0, 0 ], [ 0, 2, 1, 3, 0, 0, 1, 0, 0 ],
[ 1, 1, 0, 4, 0, 0, 1, 0, 0 ], [ 0, 3, 0, 2, 1, 0, 1, 0, 0 ],
[ 0, 2, 3, 0, 0, 1, 1, 0, 0 ], [ 1, 1, 2, 1, 0, 1, 1, 0, 0 ],
[ 2, 0, 1, 2, 0, 1, 1, 0, 0 ], [ 1, 2, 1, 0, 1, 1, 1, 0, 0 ],
[ 2, 1, 0, 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 0, 4, 1, 0, 0 ],
[ 3, 2, 0, 0, 0, 0, 2, 0, 0 ], [ 0, 0, 0, 3, 1, 0, 2, 0, 0 ],
[ 0, 0, 2, 0, 1, 1, 2, 0, 0 ], [ 1, 0, 0, 0, 2, 1, 2, 0, 0 ],
[ 2, 0, 1, 0, 0, 0, 3, 0, 0 ], [ 2, 1, 3, 0, 0, 0, 0, 1, 0 ],
[ 3, 0, 2, 1, 0, 0, 0, 1, 0 ], [ 3, 1, 1, 0, 1, 0, 0, 1, 0 ],
[ 4, 0, 0, 1, 1, 0, 0, 1, 0 ], [ 0, 0, 0, 2, 3, 0, 0, 1, 0 ],
[ 0, 1, 0, 2, 0, 2, 0, 1, 0 ], [ 1, 0, 1, 0, 0, 3, 0, 1, 0 ],
[ 0, 1, 0, 0, 0, 1, 2, 1, 0 ], [ 0, 0, 1, 0, 1, 1, 0, 2, 0 ],
[ 2, 0, 0, 0, 0, 0, 1, 2, 0 ], [ 0, 1, 2, 2, 0, 0, 0, 0, 1 ],
[ 1, 0, 1, 3, 0, 0, 0, 0, 1 ], [ 0, 2, 1, 1, 1, 0, 0, 0, 1 ],
[ 1, 1, 0, 2, 1, 0, 0, 0, 1 ], [ 0, 3, 0, 0, 2, 0, 0, 0, 1 ],
[ 1, 0, 3, 0, 0, 1, 0, 0, 1 ], [ 2, 0, 1, 0, 1, 1, 0, 0, 1 ],
[ 4, 0, 0, 0, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 2, 0, 1, 0, 1 ] ]
gap> MonotoneCatenaryDegreeOfSetOfFactorizations(Is);
4
gap> AdjacentCatenaryDegreeOfSetOfFactorizations(Is);
3
gap> EqualCatenaryDegreeOfSetOfFactorizations(Is);
4References
https://gap-packages.github.io/ numericalsgps.