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

The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is d-Koszul. It is shown that an algebra which has a reduced Gröbnerbasis that is composed of homogeneous elements of degree d is d-Koszul if and only if its associated monomial algebra is d-Koszul. The class of 2-d-determined algebras and the class 2-d-Koszul algebras are introduced. In particular, it is shown that 2-d-determined monomial algebras are 2-d-Koszul algebras and the structure of the ideal of relations of such an algebra is completely determined.