Using the linear magma model x ◇ y = ax + by
, we can produce a model of this law that contradicts Equation 3 when a+b ≠ 1
, b^2=-a/(a^2+1)
and b=(2a^2+1)/(a^5+a^3)
. This can be satisfied by taking a
to be a root of a^5+2a^4+3a^3+3a^2+a+1
. For instance, working in ℤ/pℤ
, one can take (p,a,b) = (11,1,7)
or (p,a,b) = (7,6,2)
(the latter is known to be the smallest model). See this discussion.