Numerical semigroup with Apéry set of unique expression
Definition
Let \(S\) be a numerical semigroup minimally generated by \(\{n_1, \ldots, n_e\}\) with \(n_1 < n_2 < \cdots < n_e\). It is said that \(S\) is a numerical semigroup with Apéry set of unique expression if every element in \(Ap(S, n_1)\) has unique expression, that is, \(|\mathbf{Z}(s)| = 1\) for all \(s \in Ap(S, n_1)\), where \(\mathbf{Z}(s)\) is the set of factorizations of s.
Examples
\(\circ\) Let \(S = \langle 4, 6, 11 \rangle\). It is easy to check that \(Ap(S, 4) \subseteq \{0, 4, 6, 8, 10, 11, 12, 14, 15, 16, 17\}\) and \(Ap(S, 4) = \{0, 6, 11, 17\}\). Since \(s = 6\) and \(s = 11\) are minimal generators, both have unique expression. For \(s = 17\), we have the unique expression \(17 = 6 + 11\). Therefore, \(S\) is a numerical semigroup with Apéry set of unique expression.
Examples with GAP
Nowadays, there are no functions in package NumericalSgps related to numerical semigroup with Apéry set of unique expression. However, given a numerical semigroup \(S\), the following function returns true if \(S\) is a numerical semigroup with Apéry set of unique expression and false otherwise.
gap> IsAperySetOfUniqueExpression := function(S)
> local w;
> if not IsNumericalSemigroup(S) then
> Error("The argument must be a Numerical Semigroup");
> fi;
> for w in AperyList(S) do
> if Length(Factorizations(w,S)) > 1 then
> return false;
> fi;
> od;
> return true;
> end;
function( S ) ... end\(\diamond\) Let \(S = \langle 13,14,15 \rangle\), in GAP:
gap> S := NumericalSemigroup(13,14,15);
<Numerical semigroup with 3 generators>The Apéry set of \(S\) is \(Ap(S,13) = \{0,4,5\}\) and every element (not taking into account \(w = 0\)) has unique expression, since there are minimal generators.
gap> IsAperySetOfUniqueExpression(S);
true\(\diamond\) If we consider \(S = \langle 17, 22, 24, 40 \rangle\),
gap> S := NumericalSemigroup(17, 22, 24, 40);
<Numerical semigroup with 4 generators>
gap> IsAperySetOfUniqueExpression(S);
falseFor example, \(88 \in Ap(S, 17)\) and \(88 = 4 \cdot 22 = 2 \cdot 24 + 1 \cdot 40\).
References
https://gap-packages.github.io/ numericalsgps.