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7.7 LETTERPLACE

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A Subsystem for Non-commutative Finitely Presented Associative Algebras

This section describes mathematical notions and definitions used in the subsystem of .

All algebras are assumed to be associative -algebras for being a field

or a ring .

What is and what does ?

What is ? It is a subsystem of , providing the manipulations and computations within free associative algebras over rings ,..., , where the coefficient domain is either a ring or a field, supported by .

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 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.

works with every field, supported by , 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  
7.8 Functions (letterplace)  
7.10 LETTERPLACE libraries  
7.11 Release Notes (letterplace)  


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