A monad for enumerating sets: like the list monad, but impervious to infinite descent.

A depth-first search of a data structure can fail to give a full traversal if it has an infinitely deep path. Likewise, a breadth-first search of a data structure can fall short if it has an infinitely branching node. Omega addresses this problem by using a "diagonal" traversal that gracefully dissolves such data.

So while liftM2 (,) [0..] [0..] gets "stuck" generating tuples whose first element is zero, runOmega $ liftM2 (,) (each [0..]) (each [0..]) generates all pairs of naturals.

More precisely, if x appears at a finite index in xs, and y appears at a finite index in f x, then y will appear at a finite index in each xs >>= f.

This monad gets its name because it is a monad over sets of order type omega.

Warning: Omega is only a monad when the results of runOmega are interpreted as a set; that is, a valid transformation according to the monad laws may change the order of the results. However, the same set of results will always be reachable. If you are using this as a monad, I recommendIded that you use the newer weighted-search package instead (it's also faster).