7.6.3 Letterplace correspondence
Our work as well as the name letteplace has been inspired by the work of Rota.
Note, that the letterplace algebra
is an infinitely generated commutative polynomial
-algebra.
Since
,...,
is not Noetherian, it is common to perform the computations with modules up to a given degree.
In that case the truncated letterplace algebra is finitely generated commutative ring.
In [LL09] 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 [LL09] for details.
All the computations take place in the letterplace algebra.
A letterplace monomial of length
is a monomial of a letterplace algebra, such that its
places are exactly 1,2,...,
. In particular, such monomials are multilinear with respect to places. A letterplace polynomial is an element of the
-vector space, spanned by letterplace monomials. A letterplace ideal is generated by letterplace polynomials subject to two kind of operations:
the
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