> | ## 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 53#53-algebras for 53#53 being a field 50#50 or a ring 30#30. 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 53#53 302#302,..., 303#303, where the coefficient domain 53#53 is either a ring 30#30 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 53#53[ 304#304,..., 305#305 ] and a degree (length) bound 171#171. This encodes a sub- 50#50-vector space (also called a filtered part) of 50#50 302#302,..., 303#303, spanned by all monomials of length at most 171#171. Analogously for free 53#53-subbimodules of a free 53#53-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 30#30. 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 |  |
> 7.8 Functions (letterplace) |  |
> 7.10 LETTERPLACE libraries |  |
> 7.11 Release Notes (letterplace) |  |

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