Free word problems for additive relation algebras of modules (Q1104352)

For a ring R with unit denote by \(V_ R\) the variety of additive relation algebras generated by the algebras of submodules of \(M\times M\), where M is an R-module. (The operations are the lattice meet and join, composition of relations, unary converse and relational sum: \(<a,b+c>\in f+g\) if \(<a,b>\in f\) and \(<a,c>\in g\). The variety \(V_ R\) is found to have recursively solvable word problem for free algebras in many cases, e.g., always if R is a ring with characteristic \(k\geq 1\). The classification of all the distinct varieties \(V_ R\) is given. It is proved that all the varieties \(V_ R\) are self-dual; if R is a field with characteristic zero then \(V_ R\) is not finitely axiomatizable.