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7.7 LETTERPLACE
A Subsystem for Non-commutative Finitely Presented Associative Algebras
This section describes mathematical notions and definitions used in the LETTERPLACE subsystem of SINGULAR.
All algebras are assumed to be associative
-algebras for
being a field
or a ring
.
- What is and what does LETTERPLACE?
What is LETTERPLACE? It is a subsystem of SINGULAR, providing the manipulations and computations within free associative algebras over rings
,...,
,
where the coefficient domain
is either a ring
or
a field, supported by SINGULAR.
LETTERPLACE can perform computations also in the factor-algebras of the above (via data type
qring) by two-sided ideals.Free algebras are internally represented in SINGULAR as so-called Letterplace rings.
Each such ring is constructed from a commutative ring
[
,...,
]
and a degree (length) bound
.
This encodes a sub-
-vector space (also called a filtered part) of
,...,
,
spanned by all monomials of length at most
.
Analogously for free
-subbimodules of a free
-bimodule of a fixed rank.
Within such a construction we offer the computations of Groebner (also known as Groebner-Shirshov) bases, normal forms, syzygies and many more.
We address both two-sided ideals and subbimodules of the free bimodule of the fixed rank.
A variety of monomial and module orderings is supported, including elimination orderings for both variables and bimodule components. A monomial ordering has to be a well-ordering.
LETTERPLACE works with every field, supported by SINGULAR, and with the coefficient ring
.
Note, that the elements of the coefficient field (or a ring) mutually commute with all variables.