Complete intersection lattice ideals (Q1770456)

Let \(S\) be a finitely generated, cancellative, abelian semigroup with no invertible elements, so \(S\)\ can be considered as a subsemigroup of the finitely generated abelian group \(\mathbb{Z}^{n}\oplus T\) such that \( S\cap (-S)=\{0\}\), where \(T\) is a torsion group. Let \(a_1,\dots ,a_{m}\) be a generating set for \(S\) and let \(L\) be the kernel of the group homomorphism from \(\mathbb{Z}^{m}\) to \(\mathbb{Z}^{n}\oplus T\) \ that sends the standard basis element \(e_{i}\) of \(\mathbb{Z}^{m}\) to \(a_{i}\). The lattice ideal associated to the sublattice \(L\) is the binomial ideal \( I_{L}=(\{\mathbf{x}^{\alpha ^{+}}-\mathbf{x}^{\alpha ^{-}}\mid \alpha =\alpha ^{+}-\alpha ^{_{^{-}}}\})\subset K[x_{1},\ldots ,x_{m}]\), where \(K\) is a field. The semigroup \(S\)\ is called a complete intersection if \(I_{L}\) is a complete intersection ideal. Using the notion of semigroup gluing introduced by \textit{J. C. Rosales} [Semigroup Forum 55, No. 2, 152--159 (1997; Zbl 0951.20042)] and mixed dominating matrices, the authors give a complete characterization of complete intersection lattice ideals or equivalently of finitely generated, cancellative, abelian semigroups with no invertible elements which are complete intersections. The authors also characterize set-theoretic complete intersections on binomials.