A spin system is a sequence of self-adjoint unitary operators $U_1,U_2,...$ acting on a Hilbert space $H$ which either commute or anticommute, $U_iU_j=\pm U_jU_i$ for all $i,j$; it is is called irreducible when $\{U_1,U_2,...\}$ is an irreducible set of operators. There is a unique infinite matrix $(c_{ij})$ with $0,1$ entries satisfying $$ U_iU_j=(-1)^{c_{ij}}U_jU_i, \qquad i,j=1,2,.... $$ Every matrix $(c_{ij})$ with $0,1$ entries satisfying $c_{ij}=c_{ji}$ and $c_{ii}=0$ arises from a nontrivial irreducible spin system, and there are uncountably many such matrices...

William Arveson, et al. The Structure of Spin Systems