  The state sum models in two dimensions introduced by Fukuma, Hosono and Kawai
are generalised by allowing algebraic data from a non-symmetric Frobenius
algebra. Without any further data, this leads to a state sum model on the
sphere. When the data is augmented with a crossing map, the partition function
is defined for any oriented surface with a spin structure. An algebraic
condition that is necessary for the state sum model to be sensitive to spin
structure is determined. Some examples of state sum models that distinguish
topologically-inequivalent spin structures are calculated.
