Representations of additive relation algebras by modules
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.
scientific article; zbMATH DE number 3959500
scientific article