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Bernie123

Post subject: How to define an Rorder S where S denotes a matrix algebra? Posted: Mon Jun 12, 2017 12:28 am 

Joined: Sat Jun 10, 2017 6:40 pm Posts: 4

Dear Singular Forum,
I have the following question:
Let IC denote the complex numbers.
Set R:=IC[[x]] (power series ring) and denote the onedimensional Rorder [[R, xR, xR],[x^2R, R, x^2R],[x^3R, x^3R, R]] (the latter shall be a matrix) by S.
Unforunately, I was even unable to define this Rorder S in Singular, since I only found examples of matrices with fixed entries.
So, my question is:
Is it possible to define and to do calculations with this special Rorder S in Singular?
Thanks in advance fo the help.


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hannes

Post subject: Re: How to define an Rorder S where S denotes a matrix algebra? Posted: Sat Jun 17, 2017 6:31 pm 

Joined: Wed May 25, 2005 4:16 pm Posts: 187

I do not understand that order: Is it an admissible order on the set of monomials of R? If it is, it can be represented by an integer matrix (according to lemma by Robbiano). If not, it is not usable in the context of Groebner/Standard bases.


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Bernie123

Post subject: Re: How to define an Rorder S where S denotes a matrix algebra? Posted: Sat Jun 17, 2017 8:28 pm 

Joined: Sat Jun 10, 2017 6:40 pm Posts: 4

Hi,
thank you very much for your answer. I should have added some definitions, sorry.
Let R be a Noetherian integral domain with quotient field K.
Let mod(R) denote the category of all finitely generated Rmodules.
The torsionfree modules E of mod(R) are called Rlattices. They form a subcategory lat(R).
A nonzero Ralgebra S in lat(R) is called Rorder.
Unfortunately, I am completely new to the computer algebra system Singular.
I wonder, if it is possible to do calculations with (e.g.) the previously mentioned onedimensional Rorder S in Singular.
I was unable to find examples, but then I noticed the forum.
Kind regards, Bernhard


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