results for Frobenius monads mentioned above are more general. They deal with Frobenius monads in general, and not only commutative ones. Because of that, infinite ordinals contained in $\varepsilon_0$ enter into the picture. They arise naturally in our principal model category, which bears some similarity to *2Cob*. It is a kind of planar version of *2Cob*. Something related to this model category has been described topologically in a 2-categorial context in [@KL01] (Appendix C; see also [@L08]). The infinite ordinal structure of the model category is however mentioned neither in this book, nor in the papers mentioned in the third paragraph, nor in [@K03]. In [@K03] (Section 3.6.20) we find only the vague conclusion that this ordinal structure, with which we want to deal, is “nearly about any possible drawing you can imagine”. This structure is the main novelty we obtain when we reject commutativity and pass to Frobenius monads in general.

This structure could be described by other means than by the ordinals in $\varepsilon_0$. What we need is the commutative monoid with one unary operation freely generated by the empty set of generators (see Section 6). This monoid can be isomorphically represented in the positive integers too, but 