Association - Disassociation Model

Aim: to look for symmetries in a simple system of association-disassociation.

Molecules u and v can combine to form w, which can break into u and v.

u + v <--> w

Starting from 5 w's, we have the automaton (derived from Petri net) with or without sink state.

These are aperiodic. There are no symmetry groups in the semigroup of the automata (in both cases).
Similar results hold for any initial marking of the nets (initial configuration).

However, the automaton (as a state transition graph) clearly has symmetries (like swapping input symbols t1 and t2 and reversing the order of the states). This is an example showing that the automorphism group may have elements with no connection with the automaton's transformation semigroup.