## 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.

User manual for Singular version 4.3.1, 2022, generated by texi2html.