Singular

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 some field .

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 -submodules of a free -module.

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 ideals and submodules 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.

7.7.1 Examples of use of LETTERPLACE
7.7.2 Example of use of LETTERPLACE over Z
7.7.3 Functionality and release notes of LETTERPLACE
7.7.4 References and history of LETTERPLACE