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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