I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure which I do not understand.
Suppose we have finitely presented monoid with unity $M$ with two generators say $g_1,g_2$. Lets relations for this monoid would be $Rel = {g_1^2 = e , g_2^2 = g_2 }$ where $e$ is unit element of monoid. So we have monoid $M$ to be quotient of free monoid $F$ by relations $M = F/ Rel$. $M$ is infinite. Words in $M$ has structure "$stststststst...$" etc. Rather boring ;-)
Now I want to define ring $G$ over such monoid. Lets play with field of real numbers R, as a background field. So we have ring $G = R[M]$. Suppose we are able to find such element, let's call it $g_3$ in R[M] that the following equations are satisfied:
(1) $g_i g_j = c_{ijk}g_k$ where $i,j,k=1,2,3$ just like in Lie algebra structure.
Note that $g_3$ is not element of monoid $M$ but is element of ring $G$. Also there is no antisymmetry relations for $c_{ijk}$. Then note, that from (1) we have that every element in $R[M]$ is linear combination of set of "generators" ${ e,g_i } , i=1,2,3$.
In one sentence within ring $R[M]$ we have some structure which allows us to easily compute every polynomial formula as it after some evaluations may be always turned into linear combination of generators. But such generators of $R[M]$ are different that generators of base monoid $M$.
Do You know any references where I may find examples of such structures? How they are called? They are examples of what? Are there any computer algebra systems which compute with such structures?
Additional remarks:
@ Darji - "just like Lie algebra" is about formal structure. It reminds me definition of Lie algebra, but of course $c_{ijk}$ is not antisymmetric nor Jacobi identity is satisfied so of course it is not Lie algebra.
@ Darij - Of course in general there is no associativity. In case I am interesting in this structure is associative, as it follow from simple algebra monoid which is associative, and by R[M] I mean formal combinations $sum a_i g_i$ and combinations of its multiplies as in section "two simple examples" in http://en.wikipedia.org/wiki/Group_ring
So we have noncommutative ring over monoid which is associative, has unity, and $c_{ijk}$ in j,k has both symmetric and antisymmetric components.
Further clarifications:
Structure I tried to describe consists of multiplicative monoid, and ring over it in reals. In this ring every polynomial has linear decomposition in "basis" $g_i$, somehow as in vector space. In ring every ring element allows such decompositions ( but not every linear combination of $g_i$ is ring element so it is not linear space). What is that? Do You know examples of such structures?
@Scott: You are right I am very bad English writer. Thank You for being so polite. So I will wrote it in the most explicit way I can.
I have finitely presented noncommutative monoid with unity and two generators $g_1,g_2$: M = F/Rel where $Rel = {g_1^2 = e , g_2^2 = g_2 }$, $e$ is unit element and $F$ is free monoid over two generators. Because of relations $Rel$ every elemet in monoid has form for example $g = g_1g_2g_1g_2...g_1g_2$ ( alternating finite sequence with subscripts 1212... or 2121...). Different monoid elements contains different number of multiplications. It is very simple although infinite multiplicative structure.
Then I consider monoid ring over reals $R[M]$. Every element in $R[M]$ has form:
(1) $t = r_1g_1 + r_2g_2 r_3g_1g_2+r_4g_2g_1+ r_5g_1g_2g_1 + ...+r_p g_ig_kg_i...g_s+ ...$ and so on. $r_i in R$ and $g_i in M$.
Note that in general monomial element $g_ig_k...g_s$ every subscript has value in ${1,2}$ and no two following each other subscripts are the same ( they alternate like in sequence like $1212..$ or $2121..$. Of course this is standard ring definition.
In structure, I would like to describe You here, I have strange additional property: there is element $g_3$ in ring $R[M]$ ( but it is not monoid element!) which allows following decomposition:
For every $r in R[M]$ we have
$r= r_0 e + r_1 g_1 +r_2 g_2 +r_3 g_3$
Look: there are only four terms in decomposition, even if You decompose general ring element in the form of (1). However after such decomposition I may only multiply such elements and not add them. So in fact decomposition as above, I trying to treat as some kind of "parametrization" of ring elements. Is this interesting?
As far as I know this is not standard ring property - maybe I am wrong. If I think about for example polynomial ring (that in simple case is real ring over multiplicative monoid generated by one generator $x$) such decomposition is not possible.
So I ask You if that structure was described in literature? Is it special kind of some known structure? Where to find something about it?
Thank You all for Your remarks!
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