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Decompositions of ideals into irreducible ideals in numerical semigroups. (English) Zbl 1237.20056

Summary: It is proved that each ideal \(I\) of a numerical semigroup \(S\) is in a unique way a finite irredundant intersection of irreducible ideals. The same result holds if “irreducible ideals” are replaced by “\(\mathbb{Z}\)-irreducible ideals”. The two decompositions are essentially different and, if \(n(I)\) and \(N(I)\), respectively, are the number of irreducible or \(\mathbb{Z}\)-irreducible components, it is \(n(I)\leq N(I)\leq e\), where \(e\) is the multiplicity of \(S\). However, if \(I\) is a principal ideal, then \(n(I)=N(I)=t\), where \(t\) is the type of \(S\).

MSC:

20M14 Commutative semigroups
20M12 Ideal theory for semigroups

References:

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