  We recall Borcherds's approach to vertex algebras via "singular commutative
rings", and introduce new examples of his constructions which we compare to
vertex algebras, chiral algebras, and factorization algebras. We show that all
vertex algebras (resp. chiral algebras or equivalently factorization algebras)
can be realized in these new categories $\text{VA}(A,H,S)$, but we also show
that the functors from $\text{VA}(A,H,S)$ to vertex algebras or chiral algebras
are not equivalences: a single vertex or chiral algebra may have non-equivalent
realizations as an $(A, H,S)$-vertex algebra.
