Editing Geometric algebra (section)

=== Extensions of the inner and exterior products ===
It is common practice to extend the exterior product on vectors to the entire algebra. This may be done through the use of the above-mentioned [[#Grade projection|grade projection]] operator:
: <math>C \wedge D := \sum_{r,s}\langle \langle C \rangle_r \langle D \rangle_s \rangle_{r+s} </math> &nbsp; &nbsp; (the ''exterior product'')

This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is the commutator product:
: <math>C \times D := \tfrac{1}{2}(CD-DC) </math> &nbsp; &nbsp; (the ''commutator product'')

The regressive product is the dual of the exterior product (respectively corresponding to the "meet" and "join" in this context).{{efn|[...] the exterior product operation and the join relation have essentially the same meaning. The [[Grassmann–Cayley algebra]] regards the meet relation as its counterpart and gives a unifying framework in which these two operations have equal footing [...] Grassmann himself defined the meet operation as the dual of the exterior product operation, but later mathematicians defined the meet operator independently of the exterior product through a process called [[Shuffle algebra|shuffle]], and the meet operation is termed the shuffle product. It is shown that this is an antisymmetric operation that satisfies associativity, defining an algebra in its own right. Thus, the Grassmann–Cayley algebra has two algebraic structures simultaneously: one based on the exterior product (or join), the other based on the shuffle product (or meet). Hence, the name "double algebra", and the two are shown to be dual to each other.{{sfn|ps=|Kanatani|2015|pp=112–113}}}} The dual specification of elements permits, for blades {{tmath|1= C }} and {{tmath|1= D }}, the intersection (or meet) where the duality is to be taken relative to the a blade containing both {{tmath|1= C }} and {{tmath|1= D }} (the smallest such blade being the join).{{sfn|ps=|Dorst|Lasenby|2011|p=443}}
: <math>C \vee D := ((CI^{-1}) \wedge (DI^{-1}))I </math>
with {{tmath|1= I }} the unit pseudoscalar of the algebra. The regressive product, like the exterior product, is associative.{{sfn|ps=|Vaz|da Rocha|2016|loc=§2.8}}

The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper {{Harvard citation|Dorst|2002}} gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many authors use the same symbol as for the inner product of vectors for their chosen extension (e.g. Hestenes and Perwass). No consistent notation has emerged.

Among these several different generalizations of the inner product on vectors are:
: <math> C \;\rfloor\; D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{s-r} </math> &nbsp;&nbsp;(the ''left contraction'')
: <math> C \;\lfloor\; D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{r-s} </math> &nbsp;&nbsp;(the ''right contraction'')
: <math> C * D := \sum_{r,s}\langle \langle C \rangle_r \langle D \rangle_s \rangle_{0} </math> &nbsp;&nbsp;(the ''scalar product'')
: <math> C \bullet D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{|s-r|} </math> &nbsp;&nbsp;(the "(fat) dot" product){{efn|
This should not be confused with Hestenes's irregular generalization {{tmath|1= \textstyle C \bullet_\text{H} D := \sum_{r\ne0,s\ne0}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{ \vert s-r \vert } }}, where the distinguishing notation is from {{harvp|Dorst|Fontijne|Mann|2007|p=590|loc=§B.1, which makes the point that scalar components must be handled separately with this product.}}}}

{{harvtxt|Dorst|2002}} makes an argument for the use of contractions in preference to Hestenes's inner product; they are algebraically more regular and have cleaner geometric interpretations. 
A number of identities incorporating the contractions are valid without restriction of their inputs.
For example,
: <math> C \;\rfloor\; D = ( C \wedge ( D I^{-1} ) ) I </math>
: <math> C \;\lfloor\; D = I ( ( I^{-1} C) \wedge D ) </math>
: <math> ( A \wedge B ) * C = A * ( B \;\rfloor\; C ) </math>
: <math> C * ( B \wedge A ) = ( C \;\lfloor\; B ) * A </math> 
: <math> A \;\rfloor\; ( B \;\rfloor\; C ) = ( A \wedge B ) \;\rfloor\; C </math>
: <math> ( A \;\rfloor\; B ) \;\lfloor\; C = A \;\rfloor\; ( B \;\lfloor\; C ) .</math>

Benefits of using the left contraction as an extension of the inner product on vectors include that the identity <math> ab = a \cdot b + a \wedge b </math> is extended to <math> aB = a \;\rfloor\; B + a \wedge B</math> for any vector <math>a</math> and multivector {{tmath|1= B }}, and that the [[projection (linear algebra)|projection]] operation <math> \mathcal{P}_b (a) = (a \cdot b^{-1})b </math> is extended to <math> \mathcal{P}_B (A) = (A \;\rfloor\; B^{-1}) \;\rfloor\; B</math> for any blade <math>B</math> and any multivector <math>A</math> (with a minor modification to accommodate null {{tmath|1= B }}, given [[#Projection and rejection|below]]).