  Geometric products on tensor powers $\Lambda(V)^{\otimes m}$ of an exterior
algebra and on Whitney algebras \cite{crasch} provide a rigorous version of
Grassmann's {\it regressive products} of 1844 \cite{gra1}. We study geometric
products and their relations with other classical operators on exterior
algebras, such as the Hodge $\ast-$operators and the {\it join} and {\it meet}
products in Cayley-Grassmann algebras \cite{BBR, Stew}. We establish encodings
of tensor powers $\Lambda(V)^{\otimes m}$ and of Whitney algebras $W^m(M)$ in
terms of letterplace algebras and of their geometric products in terms of
divided powers of polarization operators. We use these encodings to provide
simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras
and of two typical classes of identities in Cayley-Grassmann algebras.
