Semimodules over commutative semirings and modules over unitary commutative rings

Abstract

It is well-known that the lattice of all submodules of a module is modular. However, this is not the case for the lattice of subsemimodules of a semimodule. We show examples and describe these lattices for a given semimodule. We study closed and splitting subsemimodules and submodules of a given semimodule or module M, respectively. We derive a sufficient condition under which the lattice L c (M) of closed subsemimodules is a homomorphic image of the lattice L(M) of all subsemimodules. We describe the ordered set of splitting submodules of a module and show a natural bijective correspondence between this poset and the poset of all projections of this module. We show that this poset is orthomodular. This result extends the case known for the poset of closed subspaces of a Hilbert space which is used in the logic of quantum mechanics.