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The notion of {\it free} generalized vertex algebras is introduced. It isequivalent to the notion of {\it generalized principal subspaces} associatedwith lattices which are not necessarily integral. Combinatorial bases and thecharacters of the free generalized vertex algebras are given. As anapplication, the commutants of principal subspaces are described by usinggeneralized principal subspaces.

The Free Generalized Vertex Algebras and Generalized Principal Subspaces