Generalized Splines over $\mathbb{Z}$-Modules on Arbitrary Graphs

Let $R$ be a commutative ring with identity and $G$ a graph. An extending generalized spline on $G$ is a vertex labeling $f \in \prod_{v} M_v$, where for each edge $e=uv$ there exists an $R$-module $M_{uv}$ together with homomorphisms $\varphi_u : M_u \to M_{uv}$ and $\varphi_v : M_v \to M_{uv}$ such that $\varphi_u(f_u)=\varphi_v(f_v).$ Extending generalized splines are further generalizations for generalized splines. They can also be considered as generalized splines over modules. In this paper, we prove that some of the results for splines can be extended to generalized splines over modules $M_v=m_v\mathbb Z$ at each vertex $v$ and we define a method of a graph reduction based on graph operations on vertices and edges to produce an explicit $\mathbb{Z}$-module basis for generalized splines over modules. This corresponds to a sequence of surjective homomorphisms between the associated spline modules so that the space of splines decomposes as a direct sum of certain submodules.