Singular Manual: Letterplace correspondence

| ### 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 426#426 via the double-indexed letterplace (that is encoding the letter (= variable) and its place in the word) monomials 427#427, where 428#428 and 420#420 is the semigroup of natural numbers, starting with 1 as the first possible place. Note, that the letterplace algebra 429#429 is an infinitely generated commutative polynomial 50#50-algebra. Since 430#430 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 171#171, the truncated letterplace algebra 431#431 is finitely generated commutative polynomial 50#50-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 297#297 is a monomial of a letterplace algebra, such that its 297#297 places are exactly 1,2,..., 297#297. 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 50#50-vector space, spanned by letterplace monomials. A letterplace ideal is generated by letterplace polynomials subject to two kind of operations: the 50#50-algebra operations of the letterplace algebra and simultaneous shifting of places by any natural number 17#17. Note: Letterplace correspondence naturally extends to the correspondence over 432#432 304#304,..., 305#305 433#433, where 53#53 is a commutative unital ring. The case 434#434 is implemented, in addition to 53#53 being a field.

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