Constructing Ring over Monoid: Questions & Answers

[kakaz]

Hi!
I have following question. I will explain it with abstract notation although in fact I am working with some peculiar matrices.

I have finitely presented noncommutative monoid with identity element $I$. Presentation of this let say would be $M = <S,T;S^2>$ which means that if $S,T$ are generators of free monoid $F=<S,T>$ then $S^2 =I$ where $I$ is identity of monoid $SI=TI=IT=IS$. Then $M$ is quotient of free monoid $F$ by the given relation $M= F / [S^2-1]$.

Now I have to construct ring $R[M]$ over rationals ( complex, whatever) with structure I will build by canonical method, as a sum of elements of monoid $M$ "multiplied" by weights from field R (or even C, or whatever). Here I found that general element $Z$ of ring $R[M]$ will be defined by expression:

$Z = aI + bS +cT +dL$

where $a,b,c,d$* are in the field $R$ where $L$ is certain element which is not an element of monoid $M$ but it is properly constructed element of $R[M]$! Namely $L = [S,T] = ST - TS$. I point that monoid $M$ is multiplicative and noncommutative so $L$ is not present in monoid.

This is strange for me, and surprised me. I did not thought that it may happened: additional generator for a ring is required.

So I have situation, that monoid is generated by two generators $<S,T>$, while its ring over rationals $R[M]$ by three $<S,T,L>$ ! In fact it is even finitely presented Lie algebra for which I have structure constants computed. Presently I am looking for its matrix representations different from starting one.


This is where my knowledge ends. I am looking for some bibliography in above matter. The only things I have found was about group rings and so, then I cannot qualify if different numbers of generators is something normal or strange? Typical or interesting? Did You ever see some books or papers with other but concrete examples of such objects ( monoid rings,algebras, modules over a field )? Maybe there are even some theorems in the wild and some of You knows where may I found them?

Best regards
Kazek
* - in above term $aI$ is not needed in fact, as I have relation $S^2=I$, but it has nice shape as it is, so in this post it does not matter.

 

[JSuarez]

Why do you need L? Given Z = aI + bS + cT, if you take a = 0, b = 1 and c = 0, you get S; for a = 0, b = 0, c = 1, you get T, so ST and TS are ring elements; therefore L = ST - TS will also be a ring element. It's not necessary to include it explicitly in Z.

PS: a general element of such a ring will not have your form Z; don't forget that monoid elements of the form, for example, (T^n)(S^k) will be identified with T^n, if k is even, or (T^n)S, if k is odd.

 

[kakaz]

Why do you need L? Given Z = aI + bS + cT, if you take a = 0, b = 1 and c = 0, you get S; for a = 0, b = 0, c = 1, you get T, so ST and TS are ring elements; therefore L = ST - TS will also be a ring element. It's not necessary to include it explicitly in Z.

Good observation! Of course in typical situation, You have right: You do not need any additional element other than generators of a group. But in my situation there is a strange and interesting gain when I add L: as I wrote every element of R[M] may be expressed as :

$Z = aI + bS +cT +dL$*

I mean: every one! Every one element of my general ring R[M] is linear combination of this four ( in fact three) elements. This is very special property, so I assume it is important for structure I am trying to "algebraifying";-) That is why I am looking for help. It is not exactly canonical situation...
This also means that my finitely presented algebra over rationals for monoid R[M] has structure of vector space - this is the meaning of * equation...

PS: a general element of such a ring will not have your form Z; don't forget that monoid elements of the form, for example, (T^n)(S^k) will be identified with T^n, if k is even, or (T^n)S, if k is odd.

Exactly - but not in this case. Maybe ( which is the most probable) I do not know all equivalence relations for my monoid M. There are other ones than $S^2=I$, but I do not know them yet. I am trying to look at some similar known structures, but I cannot find. I do not think it means that they do not exists! I think that probably they are known from different descriptions etc. not exactly as abstract algebraic ones.

Thanks for Your reply: even that one was inspiration for me in some way...

 

[kakaz]

So maybe I ask once more, with better defined question: Do You know any database of finite presented structures, like algebras, groups, monoids etc. when I may look for my structure in order to check if someone else use it in other than my situation?

There are some databases in the wild: huge Cremona database of elliptic curves, Sloane database of integer sequences etc. Do You know any similar database for finitely presented algebras or groups?