This paper is intended to investigate Grassmann and Clifford algebras over
Peano spaces, introducing their respective associated extended algebras, and to
explore these concepts also from the counterspace viewpoint. The exterior
(regressive) algebra is shown to share the exterior (progressive) algebra in
the direct sum of chiral and achiral subspaces. The duality between scalars and
volume elements, respectively under the progressive and the regressive products
is shown to have chirality, in the case when the dimension n of the Peano space
is even. In other words, the counterspace volume element is shown to be a
scalar or a pseudoscalar, depending on the dimension of the vector space to be
respectively odd or even. The de Rham cochain associated with the differential
operator is constituted by a sequence of exterior algebra homogeneous subspaces
subsequently chiral and achiral. Thus we prove that the exterior algebra over
the space and the exterior algebra constructed on the counterspace are only
pseudoduals each other, when we introduce chirality. The extended Clifford
algebra is introduced in the light of the periodicity theorem of Clifford
algebras context, wherein the Clifford and extended Clifford algebras Cl(p,q)
can be embedded in Cl(p+1,q+1), which is shown to be exactly the extended
Clifford algebra. Clifford algebras are constructed over the counterspace, and
the duality between progressive and regressive products is presented using the
dual Hodge star operator. The differential and codifferential operators are
also defined for the extended exterior algebras from the regressive product
viewpoint, and it is shown they uniquely tumble right out progressive and
regressive exterior products of 1-forms.Comment: 17 pages, to appear in Adv. Appl. Clifford Algebras 16 (3) (2006