I have a basic confusion about 2D topological field theories (TFTs). In the lattice construction of 2D TFTs introduced by Fukuma et al (https://arxiv.org/abs/hep-th/9212154) only associative algebras play a role, whereas it is also understood that 2D TFTs correspond to Frobenius algebras (see e.g. http://mat.uab.es/~kock/TQFT/FS.pdf).
Frobenius algebras are associative, but also include a co-product and satisfy additional constraints e.g. the Frobenius relation. Thus, the correspondence between 2D TFT and Frobenius algebras seems more constrained than the correspondence with associative algebras.
Could someone please clarify why a Frobenius algebra does not appear in the lattice construction of Fukuma et al?