Tricksy Definitions Expressed Mathematically

Just for fun, here are a few definitions of nonstandard type to go along with those in the previous post. As any practitioner of phylogenetic nomenclature knows, most definitions are node-, branch-, or apomorphy-based, but there have been a few that don't fall into these categories.

Here are Wagner's (2004) definitions of Panbiota and Biota:

   Panbiota := (Clade ∘ prc_∩)(Homo sapiens).

   Biota := Crown(Panbiota, "extant as of or after 2004").

This is one of the few cases where it makes more sense to define the crown clade based on the total clade rather than vice versa. (Maybe the only case? Not sure.) Technically, Wagner's wording for the definition of Panbiota might be better translated as (suc_∪ ∘ min ∘ prc_∩)(Homo sapiens), but it works out to the same thing.

And here's Clarke's (2004) definition of Ichthyornis:

   Let M := "apomorphy 2" ∩ "apomorphy 5" ∩ "apomorphy 6" ∩ "apomorphy 7" ∩ "apomorphy 8".
   (These refer to apomorphies in Clarke's Ichthyornis dispar Diagnosis.)

   Ichthyornithes := Clade(YPM 1450 ← Struthio camelus ∪ Tinamus major ∪ Vultur gryphus).
   ("YPM" refers to the Yale Peabody Museum's Vertebrate Paleontology collection. YPM 1450 is the Ichthyornis dispar holotype specimen.)

   Ichthyornis := Clade((M @ YPM 1450) ∩ Ichthyornithes).