Representations of additive relation algebras by modules (Q1078216)

An additive relation algebra is a bounded modular lattice together with a monoid multiplication, a commutative monoid addition and an involution for multiplication, all satisfying natural coherence conditions. If R is a ring with unit and M an R-module, Rel(M), the submodules of \(M\times M\), form an additive relation algebra. The elementary theory of this variety of algebras is developed. Further, the main result is a representation theorem for additive relation algebras which are complemented lattices as subalgebras of Rel(R), where R is an appropriate von Neumann ring. The proof is obtained combining ''projective generator'' methods from abelian category theory with the techniques developed by von Neumann for representation of complemented modular lattices.