# association schemes, infinite schemes, semi-schemes, quasi-schemes

Some questions about possibly nonsensical ideas:

1) Can you come up with a definition of an infinite association scheme ?

2) Would infinite association schemes relate to infinite groups the way association schemes relate to finite groups ? (see http://www.ams.org/bull/2006-43-02/S0273-0979-05-01077-3/S0273-0979-05-01077-3.pdf)

3) Can you define objects that relate to semigroups and quasigroups as schemes relate to groups ?

4) What combinatorial, statistical or other properties would these infinite schemes, semi-schemes and quasi-schemes have ?

The following definition of infinite association scheme semi-rings makes sense: A set $A_i,i\in\mathcal I$ (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 $\sum_{i\in \mathcal I}A_i=J$ where $J$ denotes the infinite all $1$ matrix and $A_iA_j=\sum_{k\in\mathcal I}\gamma_{i,j}^kA_k$ with $\gamma_{i,j}^k$ in $\mathbb N\cup\lbrace\infty\rbrace$ 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 $\sum_{i\in\mathcal I}\lambda_i A_i$ 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.