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We give a characterization of hypercyclic finitely generated abelian semigroups of matrices on $$\mathbb{C }^{n}$$ using the extended limit sets (the J-sets). Moreover we construct for any $$n\ge 2$$ an abelian semigroup $$G$$ of GL $$(n, \mathbb{C })$$ generated by $$n+1$$ diagonal matrices which is locally hypercyclic but not hypercyclic and such that J $$_{G}(e_{k}) = \mathbb{C }^{n}$$ for every $$k= 1,\dots , n$$, where $$(e_{1},\dots , e_{n})$$ is the canonical basis of $$\mathbb{C }^{n}$$. This gives a negative answer to a question raised by Costakis and Manoussos. [ABSTRACT FROM AUTHOR]