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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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