The species X of singletons. Puts a singleton structure on an underlying label set of size 1, and no structures on any other underlying label sets. x is also provided as a synonym.

The species E of sets. Puts a singleton structure on any underlying label set.

The species C of cyclical orderings (cycles/rings).

The species of bracelets (i.e. cycles that can also be flipped).

The species L of linear orderings (lists). Since linear orderings are isomorphic to cyclic orderings with a hole, we may take linOrd = oneHole cycle as the default implementation; linOrd is included in the Species class so it can be special-cased for enumeration.

The species p of subsets is given by subset = set * set. subset is included in the Species class so it can be overridden when enumerating structures: by default the enumeration code would generate a pair of the subset and its complement, but normally when thinking about subsets we only want to see the elements in the subset. To explicitly enumerate subset/complement pairs, you can use set * set directly.

Subsets of size exactly k, ksubset k = (set `ofSizeExactly` k) * set. Included with a default definition in the Species class for the same reason as subset.

Structures of the species e of elements are just elements of the underlying set, element = singleton * set. Included with a default definition in Species class for the same reason as subset and ksubset.

Partitional composition. To form all (f `o` g)-structures on the underlying label set U, first form all set partitions of U; for each partition p, put an f-structure on the classes of p, and a separate g-structure on the elements in each class.

Cartisian product of two species. An (f >< g)-structure consists of an f-structure superimposed on a g-structure over the same underlying set.

Functor composition of two species. An (f @@ g)-structure consists of an f-structure on the set of all g-structures.

Only put a structure on underlying sets whose size satisfies the predicate.

Only put a structure on underlying sets of the given size. A default implementation of ofSize (==k) is provided, but this method is included in the Species class as a special case since it can be more efficient: we get to turn infinite lists of coefficients into finite ones.

Don't put a structure on the empty set. The default definition uses ofSize; included in the Species class so it can be overriden in special cases (such as when reifying species expressions).

rec f is the least fixpoint of (the interpretation of) the higher-order species constructor f.

Omega is the pseudo-species which only puts a structure on infinite label sets. Of course this is not really a species, but it is sometimes a convenient fiction to use Omega to stand in for recursive occurrences of a species.