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