An Algebraic Description of Development of Hierarchy

We propose an algebraic description of emergence of new levels in trophic level networks. Trophic level networks are described by directed graphs. Their properties are surveyed in terms of an adjunction on a subcategory of the category of directed graphs. In particular, it is shown that trophic level networks are invariant under the composition of the right adjoin functor and the left adjoin functor. This invariance of trophic level networks can be broken by introducing the notion of time into the left adjoint functor. This leads to changes in trophic level networks. We show that the left adjoin functor consists of an intra-level process and an inter-level process. An inconsistency between them arises by the introduction of time. Negotiation between the intra-level process and the inter-level process can resolve the inconsistency at a level, however, a new inconsistency can arises at an emerged new level. Thus our algebraic description can follow indefinite development of trophic hierarchy.