A function defined on an arbitrary minor-closed class of matroids is a "Tutte function" if it satisfies the parametrized deletion-contraction law
F(M) = de F(M\e) + ce F(M/e)
whenever e is a point of M that is neither a loop nor a coloop. F need not have any other Tutte-style properties like multiplicativity. Here de and ce are constants associated with e, independent of M but depending on the point e.
Functions of this kind appear in statistical physics and knot theory.
Joanna Ellis-Monaghan and I are studying the modules and algebras behind Tutte functions.