[More information in EDIT 2] If one defines an operation $\odot: V\times V\rightarrow V$ between the elements of a linear vector space, what properties should this operation have in order for a "$\odot$-exponential" function to be defined: $$ \overset{\odot}{\exp}(A):=\sum_{k=0}^{\infty}\frac{A^{\odot k}}{k!} $$ such that a property like $$ \overset{\odot}{\exp}(A)\odot\overset{\odot}{\exp}(B)=\overset{\odot}{\exp}(A+B) $$ holds true?
EDIT 2: It's best if I tell the whole story. I am thinking of a linear space of lists. Each list looks like $[a_i,b_j,\dots]$, where the indices are positive integers. The elements of a list commute if they have different indices, so that $[a_1,b_1]\neq[b_1,a_1]$, but $[a_1,b_2]=[b_2,a_1]$. The operation $\odot$ joins lists, e.g. $$[a_1,b_1,c_3]\odot[c_3,d_1]=[a_1,b_1,c_3,c_3,d_1]$$ Also, $\odot$ is bilinear, e.g. $$(\alpha[a_1]+\beta[a_2])\odot(\gamma[b_3])=\alpha\gamma[a_1,b_3]+\beta\gamma[a_2,b_3]$$ And for $A=[a_i,b_j,\dots]$ I would define $A^{\odot 0}=[]$ (the empty list) and $A^{\odot 1}=A$.
I find myself dealing with an expression that looks identical to the definition of the usual exponential. How far can I take it with the analogies?