Embedding distributions in algebras of generalized functions with singularities

It is a known open question whether in Rosinger's nowhere dense quotient algebra distributions can be embedded, ensuring consistency with partial derivatives. In this paper, we construct algebras of generalized functions with nowhere dense singularities similar to Rosinger's algebra in which such an embedding of distributions is possible. Moreover, in some of the algebras, the embedding preserves the products of smooth functions, Further, we indicate that most of the properties of Rosinger's algebra, in particular concerning the solvability of nonlinear PDEs, remain valid in the new algebras. Summarizing, we give a positive answer to the above-mentioned open question, if one allows a modification in the definition of the algebra that does not affect its basic properties.