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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.

Already Feynman and Rota encoded the monomials (words) of the free algebra $x_{i_1} x_{i_2} \dots x_{i_m} \in K\langle x_1,\ldots,x_n \rangle$ via the double-indexed letterplace (that is encoding the letter (= variable) and its place in the word) monomials $x(i_1 \vert 1) x(i_2 \vert 2) \dots x(i_m \vert m) \in K[X\times N]$, where $X=\{x_1,\ldots,x_n\}$ and $N$ is the monoid of natural numbers, starting with 0 which cannot be used as a place.

Note, that the letterplace algebra $K[X \times N]$ is an infinitely generated commutative polynomial $K$-algebra. Since $K<$ $x_1$,..., $x_n$ $>$ 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 $m$ is a monomial of a letterplace algebra, such that its $m$ places are exactly 1,2,..., $m$. In particular, such monomials are multilinear with respect to places. A letterplace polynomial is an element of the $K$-vector space, spanned by letterplace monomials. A letterplace ideal is generated by letterplace polynomials subject to two kind of operations:

the $K$-algebra operations of the letterplace algebra and simultaneous shifting of places by any natural number $n$.
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