### 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 via the double-indexed letterplace (that is encoding the letter (= variable) and its place in the word) monomials , where and is the semigroup of natural numbers, starting with 1 as the first possible place. Note, that the letterplace algebra is an infinitely generated commutative polynomial -algebra. Since 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 , the truncated letterplace algebra is finitely generated commutative polynomial -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 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 (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 -vector space, spanned by letterplace monomials. A letterplace ideal is generated by letterplace polynomials subject to two kind of operations:

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

Note: Letterplace correspondence naturally extends to the correspondence over

,..., , where is a commutative unital ring. The case is implemented, in addition to

being a field.

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