On characterization of finite modules by hypergraphs

With a finite R-module M we associate a hypergraph 𝒞𝒥ℋ_R(M) having the set V of vertices being the set of all nontrivial submodules of M. Moreover, a subset E_i of V with at least two elements is a hyperedge if for K, L in E_i there is K ∩ L ≠ = 0 and E_i is maximal with respect to this property. We investigate some general properties of 𝒞𝒥ℋ_R(M), providing condition under which 𝒞𝒥ℋ_R(M) is connected, and find its diameter. Besides, we study the form of the hypergraph 𝒞𝒥ℋ_R(M) when M is semisimple, uniform module and it is a direct sum of its each two nontrivial submodules. Moreover, we characterize finite modules with three nontrivial submodules according to their co-intersection hypergraphs. Finally, we present some illustrative examples for 𝒞𝒥ℋ_R(M).