Editing Combinatorial species (section)

===Multiplication ===
'''Multiplying''' species is slightly more complicated. It is possible to just take the Cartesian product of sets as the definition, but the combinatorial interpretation of this is not quite right. (See below for the use of this kind of product.) Rather than putting together two unrelated structures on the same set, the [[multiplication operator]] uses the idea of splitting the set into two components, constructing an ''F''-structure on one and a ''G''-structure on the other.<ref>{{harvnb|Joyal|1981|loc=§ 2.1. Definition 5}}</ref>
:<math>(F \cdot G)[A] = \sum_{A=B+C} F[B] \times G[C].</math>
This is a disjoint union over all possible binary partitions of&nbsp;''A''. It is straightforward to show that multiplication is [[Associativity|associative]] and [[Commutativity|commutative]] ([[up to]] [[isomorphism]]), and [[Distributivity|distributive]] over addition. As for the generating series, (''F''&nbsp;·&nbsp;''G'')(''x'')&nbsp;=&nbsp;''F''(''x'')''G''(''x'').

The diagram below shows one possible (''F''&nbsp;·&nbsp;''G'')-structure on a set with five elements. The ''F''-structure (red) picks up three elements of the base set, and the ''G''-structure (light blue) takes the rest. Other structures will have ''F'' and ''G'' splitting the set in a different way. The set (''F''&nbsp;·&nbsp;''G'')[''A''], where ''A'' is the base set, is the disjoint union of all such structures.

[[File:Multiplication of combinatorial species.svg|200px|center]]

The addition and multiplication of species are the most comprehensive expression of the sum and product rules of counting.{{Citation needed|reason=In what sense are they "the most comprehensive"? is there a citation with more context?|date=July 2024}}