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The relationship between an algebra and its associated monomial algebra isinvestigated when at least one of the algebras is $d$-Koszul. It is shown thatan algebra which has a reduced \grb basis that is composed of homogeneouselements of degree $d$ is $d$-Koszul if and only if its associated monomialalgebra is $d$-Koszul. The class of 2-$d$-determined algebras and the class2-$d$-Koszul algebras are introduced. In particular, it shown that2-$d$-determined monomial algebras are 2-$d$-Koszul algebras and the structureof the ideal of relations of such an algebra is completely determined.

$d$-Koszul algebras, 2-$d$ determined algebras and 2-$d$-Koszul algebras