co.combinatorics - association schemes, infinite schemes, semi-schemes, quasi-schemes

A (probably not very brilliant) partial answer to the first question.

The following definition of infinite association scheme semi-rings makes sense: A set $A_i,iinmathcal I$ (with $mathcal I$ not necessarily finite) of infinite matrices (indexed by $mathbb N$ or $mathbb Z$) with coefficients in ${0,1}$ containing the infinite identity matrix such that $sum_{iin mathcal I}A_i=J$ where $J$ denotes the infinite all $1$ matrix and
$A_iA_j=sum_{kinmathcal I}gamma_{i,j}^kA_k$ with $gamma_{i,j}^k$ in
$mathbb Ncuplbraceinftyrbrace$ and where the last sum is finite.
We require moreover the equalities $gamma_{i,j}^k=gamma_{j,i}^k$.
All operations are then well-defined on
the semiring $sum_{iinmathcal I}lambda_i A_i$ of finite sums with coefficients in
$mathbb R_{geq 0}cup{infty}$ and can even be extended to infinite sums (this is
useful since the Hadamard product identity, $J$ is an infinite sum).
Negative coefficients should be avoided.

I am not convinced of the interest of such a structure.