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 : $$(\exists k)(\forall m \in M)[m^k = m^{k+1}],$$ 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.

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.

Hi there everyone,

I'm in the process of learning about periodic languages, and I'm really confused over the vocabulary used to describe the periodicity of (the 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 : $$(\exists k)(\forall m \in M)[m^k = m^{k+1}],$$ 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 monoid and semigroups?

Thank you.

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 : $$(\exists k)(\forall m \in M)[m^k = m^{k+1}],$$ 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.