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7.6.3 Letterplace correspondence
We utilize the ideas of Feynmann and Rota and encode
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).
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].
All the computations take place in the latter algebra. A letterplace ideal
is a subset of a special vector space
, which is spanned by all letterplace
monomials. A letterplace monomial of length m is a monomial of
a letterplace algebra, such that its m places are exactly 1,2,...,
.
That is a multilinearity with respect to places occur.
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