Let \({{\mathcal {S}}}\subseteq {{\mathbb {Z}}}^m \oplus T\) be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in \({\mathcal {S}}\) having at least two factorizations of the same length, namely the ideal \({\mathcal {L}}_{{\mathcal {S}}}\). To this end, we work with a certain (lattice) ideal associated to the monoid \({\mathcal {S}}\). Our study can be seen as a new approach generalizing [9], which only studies the case of numerical semigroups. When \({{\mathcal {S}}}\) is a numerical semigroup we give three main results: (1) we compute explicitly a set of generators of the ideal \({\mathcal {L}}_{\mathcal S}\) when \({\mathcal {S}}\) is minimally generated by an almost arithmetic sequence; (2) we provide an infinite family of numerical semigroups such that \({\mathcal {L}}_{{\mathcal {S}}}\) is a principal ideal; (3) we classify the computational problem of determining the largest integer not in \({\mathcal {L}}_{{\mathcal {S}}}\) as an \(\mathcal {NP}\)-hard problem.

令\（{{\ mathcal {S}}} \ subseteq {{\ mathbb {Z}}} ^ m \ oplus T \）是有限生成和简化的类半体。在本文中，我们开发了一种通用策略来研究\（{\ mathcal {S}} \）中具有至少两个相同长度分解因子的元素集，即理想\（{\ mathcal {L}} _ { {\ mathcal {S}}} \）。为此，我们要使用与等式\（{\ mathcal {S}} \）相关联的某个（晶格）理想。我们的研究可以看作是一种推广的新方法[9]，该方法仅研究数值半群的情况。当\（{{\ mathcal {S}}} \}是一个数值半群时，我们给出三个主要结果：（1）我们显式计算出一组理想\（{\ mathcal {L}} _ {\ mathcal S} \）当\（{\ mathcal {S}} \\）由几乎是算术序列最小生成时；（2）我们提供了一个无限的数值半群族，使得\（{\ mathcal {L}} _ {{\ mathcal {S}}} \\}是一个主要理想；（3）我们将确定不在\（{\ mathcal {L}} _ {{\ mathcal {S}}} \}中的最大整数的计算问题归类为\（\ mathcal {NP} \）- hard问题。