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I'm learning about periodic languages, and I'm confused over the vocabulary used to describe the periodicity of (syntactic) monoids.

If I understand correctly, a monoid M is periodic if : $$(\forall m \in M)(\exists i \neq j)[m^i = m^j],$$ and it is aperiodic if : $$(\forall m \in M)(\exists k)[m^k = m^{k+1}],$$ and then an aperiodic monoid is periodic. Where does that bizarre vocabulary come from?

And in the same vein, what would be the book you'd recommend on monoids and semigroups?

Thank you.

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  • $\begingroup$ Nomenclature is seldom optimal! We also have irreducible but non-reduced schemes, and other niceties. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jul 17, 2010 at 3:00
  • $\begingroup$ In the definition of aperiodicity, you had (I corrected it) inverted the quantifiers. This was harmless for finite monoids but not for infinite ones. $\endgroup$
    – Duchamp Gérard H. E.
    Commented Sep 18, 2015 at 13:28

1 Answer 1

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The notion "periodic monoid" comes from the notion "periodic element", that is an element $a$, such that the sequence $a,a^2,a^3,\dots $ is eventually periodic. In particular every finite monoid is periodic. The notion "aperiodic monoid" (usually in this case we are talking about finite monoids) means "all subgroups are trivial", that is in the Crohn-Rhodes decomposition all "cycles" are trivial. One of the best books about (finite) monoids and their applications to automata and languages is Eilenberg, Automata, languages, and machines. Vol. A., B. For abstract semigroups, there is still the classic book by Clifford and Preston, "The algebraic theory of semigroups", and more recent books of Howie, "Fundamentals of semigroup theory" and Lawson, "Inverse semigroups. The theory of partial symmetries" among many others.

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