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7.6.3 Letterplace correspondence
Already Feynman and Rota encoded
the monomials (words) of the free algebra
via the double-indexed letterplace
(that is encoding the letter (= variable) and its place in the word) monomials
, where
and is the monoid of natural numbers, starting with 0 which cannot be used as a place.
Note, that the latter letterplace algebra
is an infinitely generated commutative polynomial algebra.
Since
,...,
is not Noetherian, it is common to perform the computations up to a given degree.
In that case the truncated letterplace algebra is (a) finitely generated (commutative ring).
In [LL] a natural shifting on letterplace polynomials was introduced and used.
Indeed, there is 1-to-1 correspondence between graded two-sided ideals
of a free algebra and so-called letterplace ideals in the letterplace algebra, see [LL] for details.
All the computations take place in the latter algebra.
A letterplace monomial of length
is a monomial of a letterplace algebra, such that its
places are exactly 1,2,...,
. That is, such monomials are multilinear with respect to places. A letterplace ideal is generated by letterplace polynomials, which
start from the place 1 (i.e. elements from the special vector space
) subject to two kind of operations: the
-algebra operations of the letterplace algebra and simultaneous shifting of places by any natural number .
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