A nonpositive curvature property of modular semilattices

The orthoscheme complex of a graded poset is metrization its order such that the simplex each maximal chain isometric to Euclidean vertices $$0, e_1,e_1+e_2,\ldots , e_1+e_2+ \cdots + e_n$$ . This notion was introduced by Brady and McCammond in geometric group theory, has applications discrete optimization submodularity theory. We address question which can yield with CAT(0) property. modular lattice been shown be CAT(0), it conjectured this case for semilattice. In paper, we prove conjecture affirmatively. result implies larger class weakly graphs yields complexes.