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7.9.5 Letterplace correspondence

The name letteplace has been inspired by the work of Rota and, independently, Feynman.

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 semigroup of natural numbers, starting with 1 as the first possible place. Note, that the letterplace algebra $K[X \times N]$ is an infinitely generated commutative polynomial $K$-algebra. Since $K\langle x_1,\ldots,x_n \rangle$ is not Noetherian, it is common to perform the computations with its ideals and modules up to a given degree bound.

Subject to the given degree (length) bound $d$, the truncated letterplace algebra $K[X\times (1, ..., d)]$ is finitely generated commutative polynomial $K$-algebra.

In [LL09] a natural shifting on letterplace polynomials was introduced and used. Indeed, there is 1-to-1 correspondence between two-sided ideals of a free algebra and so-called letterplace ideals in the letterplace algebra, see [LL09], [LL13], [LSS13] and [L14] for details. Note, that first this correspondence was established for graded ideals, but holds more generally for arbitrary ideals and subbimodules of a free bimodule of a finite rank. All the computations internally 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 (i.e. no place, smaller than the length is omitted or filled more than with one letter). 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 $n$.

Note: Letterplace correspondence naturally extends to the correspondence over

$R<$ $x_1$,..., $x_n$ $>$, where $R$ is a commutative unital ring. The case $R=Z$ is implemented, in addition to

$R$ being a field.

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