What's the order of a semigroup?

Question

For a group the order, of an element is the smallest positive integer m such that a^m = e. But what's the order of an element of a semigroup? Or there isn't anything like that?

Answer 1

You can define the period of an element $a$ : the smallest integer $n>0$ such that there is $k\in \mathbb{N}$ with $a^{k+n} = a^k$.

Of course like the (finite) order of an element it may not exist (the element is aperiodic and generates an infinite sub-semigroup) ; and for a group it coincides with the order of the element.

Answer 2

The following definitions were given in the reference book [1], p. 19 and seem to have been accepted since then:

The order of an element $a$ is the cardinal of the semigroup $\langle a \rangle$ generated by $a$. If $\langle a \rangle$ is finite, there exist integers $i, p > 0$ such that $a^{i+p} = a^i$. The minimal $i$ and $p$ with this property are called respectively the index and the period of $a$.

[1] A.H. Clifford, G.B. Preston, The algebraic theory of semigroups Vol I. Mathematical Surveys, vol. 7. American Mathematical Society, Providence (1961)