  Well-known operations defined on a non-degenerate inner product vector space
are extended to the case of a degenerate inner product. The main obstructions
to the extension of these operations to the degenerate case are (1) the index
lowering operation is not invertible, and (2) we cannot associate to the inner
product in a canonical way a reciprocal inner product on the dual of the vector
space. This article shows how these obstructions can be avoided naturally,
allowing a canonical definition of covariant contraction for some important
special cases.
  The primary motivation of this article is to lay down the algebraic
foundation for the construction of invariants in Singular Semi-Riemannian
Geometry, especially those related to the curvature. It turns out that the
operations discussed here are enough for this purpose (arXiv:1105.0201,
arXiv:1105.3404, arXiv:1111.0646). Such invariants can be applied to the study
of singularities in the theory of General Relativity (arXiv:1111.4837,
arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099, arXiv:1112.4508).
