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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 $K$-algebras for some field $K$.

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 $R$ $<x_1$,..., $x_n >$, where the coefficient domain $R$ is either a ring $Z$ 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 $R$[ $x_1$,..., $x_n$ ] and a degree (length) bound $d$.

This encodes a sub- $K$-vector space (also called a filtered part) of $K$ $<x_1$,..., $x_n >$, spanned by all monomials of length at most $d$. Analogously for free $R$-submodules of a free $R$-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 $Z$. 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