We establish, for finitely generated abelian semigroups G of matrices on ℝn, and by using the extended limit sets (the J-sets), the following equivalence analogous to the complex case: (i) G is hypercyclic, (ii) JG(vη) = ℝn for some vector vη given by the structure of G, (iii) G(vη) = ℝn. This answer a question raised by the author. Moreover we construct for any n = 2 an abelian semigroup G of GL(n, ℝ) generated by n + 1 diagonal matrices which is locally hypercyclic (or J-class) but not hypercyclic and such that JG(ek) = ℝn for every k = 1,…, n, where (e1,…, en) is the canonical basis of ℝn. This gives a negative answer to a question raised by Costakis and Manoussos