Algebraic Emergence

The concept of emergence has been applied to a wide variety of systems and it is frequently mentioned in the context of artificial life and intelligence. Some formulations exist, but there is no generally accepted mathematical theory of emergence. Here, we fix the context to mathematical models of discrete dynamical systems (semigroupoids in general) to obtain clarity about emergent properties. These algebraic structures have great modeling power including many artificial life systems. At the same time, they allow us to formulate precise conditions for emergence. First, there should be a morphic relation between the underlying and the emergent systems, otherwise there is no causal or predictive connection. Second, emergence happens when a quotient structure (coarse-grained dynamics) is not a substructure of the original system. We survey small groups to show that algebraic emergence is neither ubiquitous nor rare. Finally, we discuss connections with hierarchical decompositions.