Homogeneous modular lattices are distributive (Q304193)

A structure \(A\) is homogeneous if any partial isomorphism between finitely generated substructures can be extended to an automorphism of the structure \(A\). \textit{A. Abogatma} and \textit{J. K. Truss} [Order 32, No. 2, 239--243 (2015; Zbl 1348.06004)] have constructed uncountably many contable homogeneous lattices. They conjectured that any modular homogeneous lattice is distributive. Here, the author shows that this conjecture is true. In fact, he shows that a very weak form of homogeneity suffices to show that a modular lattice is distributive.