CUBICAL SPECIES AND NONASSOCIATIVE-ALGEBRAS

We lay down the foundations of a theory of cubical species, as a varia nt of Joyal's classical theory of species (A. Joyal, Adv. Math. 42 (19 81),:1-82). Informally, a cubical species associates in a functorial w ay a set of structures to each hypercube. In this context, the hyperoc tahedral groups replace the symmetric soups. We analyze cubical specie s, molecular cubical species,;and basic operations among them, along w ith explicit examples. We show, in particular, that the cubical produc t gives rise, in a natural way, to a commutative nonassociative ring o f formal power series. We conclude with a detailed analysis of this no nassociative ring. (C) 1998 Academic Press.