Specieslike clusters based on identical ancestor points

We introduce several axioms which may or may not hold for any given subgraph of the directed graph of all organisms (past, present and future) where edges represent biological parenthood, with the simplifying background assumption that life does not go extinct. We argue these axioms are plausible for species: if one were to define species based purely on genealogical relationships, it would be reasonable to define them in such a way as to satisfy these axioms. The main axiom we introduce, which we call the identical ancestor point axiom, states that for any organism in any species, either the species contains at most finitely many descendants of that organism, or else the species contains at most finitely many non-descendants of that organism. We show that this (together with a convexity axiom) reduces the subjectivity of species, in a technical sense. We call connected sets satisfying these two axioms "specieslike clusters." We consider the question of identifying a set of biologically plausible constraints that would guarantee every organism inhabits a maximal specieslike cluster subject to those constraints. We provide one such set consisting of two constraints and show that no proper subset thereof suffices.