Editing Combinatorial species (section)

===Further operations===
There are a variety of other manipulations which may be performed on species. These are necessary to express more complicated structures, such as [[directed graph]]s or [[bigraph]]s.

'''Pointing''' selects a single element in a structure.<ref>{{cite book |last1=Flajolet |first1=Philippe |author-link1=Philippe Flajolet |last2=Sedgewick |first2=Robert |author-link2=Robert Sedgewick (computer scientist) |date=2009 |title=Analytic combinatorics}}</ref> Given a species ''F'', the corresponding pointed species ''F''<sup>•</sup> is defined by ''F''<sup>•</sup>[''A''] = ''A'' &times; ''F''[''A'']. Thus each ''F''<sup>•</sup>-structure is an ''F''-structure with one element distinguished. Pointing is related to [[differentiation (mathematics)|differentiation]] by the relation ''F''<sup>•</sup> = ''X''·''F' '', so ''F''<sup>•</sup>(''x'') = ''x'' ''F' ''(''x''). The species of [[pointed set]]s, ''E''<sup>•</sup>, is particularly important as a building block for many of the more complex constructions.

The '''Cartesian product''' of two species{{citation needed|date=December 2018}} is a species which can build two structures on the same set at the same time. It is different from the ordinary multiplication operator in that all elements of the base set are shared between the two structures. An (''F'' &times; ''G'')-structure can be seen as a superposition of an ''F''-structure and a ''G''-structure. Bigraphs could be described as the superposition of a graph and a set of trees: each node of the bigraph is part of a graph, and at the same time part of some tree that describes how nodes are nested. The generating function (''F'' &times; ''G'')(''x'') is the Hadamard or coefficient-wise product of ''F''(''x'') and ''G''(''x'').

The species ''E''<sup>•</sup> &times; ''E''<sup>•</sup> can be seen as making two independent selections from the base set. The two points might coincide, unlike in ''X''·''X''·''E'', where they are forced to be different.

As functors, species ''F'' and ''G'' may be combined by '''functorial composition''':{{citation needed|date=December 2018}} <math>(F \,\Box\, G) [A] = F[G[A] ]</math> (the box symbol is used, because the circle is already in use for substitution). This constructs an ''F''-structure on the set of all ''G''-structures on the set ''A''. For example, if ''F'' is the functor taking a set to its power set, a structure of the composed species is some subset of the ''G''-structures on ''A''. If we now take ''G'' to be ''E''<sup>•</sup> &times; ''E''<sup>•</sup> from above, we obtain the species of directed graphs, with self-loops permitted. (A directed graph is a set of edges, and edges are pairs of nodes: so a graph is a subset of the set of pairs of elements of the node set ''A''.) Other families of graphs, as well as many other structures, can be defined in this way.