For a real Banach lattice \(E\), let \textit{L}\(^r(E)\) denote all the regular linear operators on \(E\) (i.e., those that are the difference of positive operators) which has a natural order and algebra structure. For Banach lattices \(E\) and \(F\), it is established that if \(\Phi\) is an order and algebra isomorphism from \textit{L}\(^r(E)\) onto \textit{L}\(^r(F)\), then there is a linear order isomorphism \(U\) from \(E\) onto \(F\) inducing the map \(\Phi\). In particular, \(\Phi(T)= UTU^{-1} \) for each \(T \in\) \textit{L}\(^r(E)\). The proof utilizes the tensor products of vector lattices. This result is analagous to the known result for real Banach spaces, namely that an algebra isomorphism between the spaces of linear operators on two Banach spaces is induced by a linear isomorphism between the Banach spaces.