Let $S$ be a finite semigroup. I want to define an exponent of $S$ which is a generalization of the familiar term in finite groups.
Recall that every element $a\in S$ determines a unique pair of positive integers $\iota=\mathrm{ind}(a)$ and $\rho=\mathrm{prd}(a)$, called the index of $a$ and the period of $a$, respectively. These are the smallest positive integers such that $a^{\iota}=a^{\iota+\rho}$.
I have two options to define $\exp(S)$:
(a) Given an element $a\in S$, it can be shown that the Kernel of $a$, namely the set $$ K_a=\{a^{\iota},a^{\iota+1},\ldots,a^{\iota+\rho-1}\} $$ forms a cyclic group. Hence, one of the powers of $a$, say $a^k$, plays the rule of the identity element, so $(a^k)^2=a^k$. Hence, for every element $a\in S$, there exists a positive integer $k$ such that $a^k$ is idempotent. Thus, it reasonable to define the exponent of a semigroup $S$ to be the smallest integer $n$ such that $s^n$ is idempotent for all $s\in S$.
(b) If $S=\{s_1,\ldots,s_m\}$, then it is also reasonable to define
$$
\exp(S)=\mathrm{lcm}(\mathrm{prd}(s1),\ldots,\mathrm{prd}(s_m))
$$
Note that if $I=\max\{\mathrm{ind(s_1)},\ldots,\mathrm{ind(s_m)}\}$ and $I\leq\exp(S)$, then $a^{\exp(S)}$ is idepotent.
Is the two definitions are equivalent in general? If yes, what are the reasoning? If not, what is the "right" way to define $\exp(S)$?
I use the first definition for the exponent of a finite semigroup.
I would rather call period of $S$ the lcm of the periods of the elements of $S$ and index of $S$ the maximum of the indices of the elements of $S$. Indeed, if $i$ is the index of $S$ and $p$ is the period of $S$ in the previous sense, then $S$ satifies the identity $x^i = x^{i+p}$.
The two notions usually do not coincide. In particular, if $S$ is an aperiodic finite semigroup, then its period is $1$ by definition, but its exponent can be anything you want.