Symmetries of Automata

Attila Egri-Nagy, C.L. Nehaniv

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For a given reachable automaton A, we prove that the (state-)endomorphism monoid End(A) divides its characteristic monoid M(A). Hence so does its (state-)automorphism group Aut(A), and, for finite A, Aut(A) is a homomorphic image of a subgroup of the characteristic monoid. It follows that in the presence of a (state-) automorphism group G of A, a finite automaton A (and its transformation monoid) always has a decomposition as a divisor of the wreath product of two transformation semigroups whose semigroups are divisors of M(A), namely the symmetry group G and the quotient of M(A) induced by the action of G. Moreover, this division is an embedding if M(A) is transitive on states of A. For more general automorphisms, which may be non-trivial on input letters, counterexamples show that they need not be induced by any corresponding characteristic monoid element.
Original languageEnglish
Pages (from-to)48-57
Number of pages10
JournalAlgebra and Discrete Mathematics
Issue number1
Publication statusPublished - 24 Mar 2015


  • 2010 Mathematics Subject Classification: 20B25, 20E22, 20M20, 20M35, 68Q70.


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