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.
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.
A theorem of A. C. Fleck (J. ACM 12(4):566-569, 1965) implies that group of *state* automorphisms (i.e. one that is allowed to permute states, but not inputs) divides the semigroup of the automaton. However, the order 2 automorphism we mentioned above interchanges the labels. In this case, where the semigroup of the automaton has only trivial subgroups, Fleck's theorem implies there are no state automorphisms (except the identity).
ReplyDeleteBy considering in an automorphism that the two states with only two outgoing arrows must be permuted, it easily follows that the order 2 symmetry is the only non-trivial symmetry of this automaton.
ReplyDeleteSimilarly, the same holds for the automaton generated for any initial distribution molecules: it is aperiodic, but has exactly one non-trivial order two symmetry like the one above.