Sections in orthomodular structures of decompositions

There is a family of constructions to produce orthomodular structures from modular lattices, lattices that are M and M*-symmetric, relation algebras, the idempotents of a ring, the direct product decompositions of a set or group or topological space, and from the binary direct product decompositions of an object in a suitable type of category. We show that an interval [0, a] of such an orthomodular structure constructed from A is again an orthomodular structure constructed from some B built from A. When A is a modular lattice, this B is an interval of A, and when A is an object in a category, this B is a factor of A.