  In this paper, we introduce the class of finitely semi-graded algebras which
extends the connected graded algebras finitely generated in degree one. The
Koszul behavior of finitely semi-graded algebras is investigated by the
distributivity of some associated lattice of ideals. The Hilbert series, the
Poincar\'e series and the Yoneda algebra are defined for this class of
algebras. Finitely semi-graded algebras include many important examples of non
$\mathbb{N}$-graded algebras finitely generated in degree one coming from
mathematical physics, and for these concrete examples the Koszulity will be
established, as well as, the explicit computation of its Hilbert and Poincar\'e
series.
