Exponent Laws for Natural Powers in a Monoid

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(M,\cdot)$ be a monoid with identity $e$, products written by juxtaposition as in Semigroups (i), let $x,y\in M$ and $a,b\in\mathbb{N}_{0}$, and let $x^{a}$ be the power of $x$, with $x^{0}=e$ by Powers with Exponents in the Natural Numbers with Zero (ii).

(i) Successor.

$$x^{a+1}=x^{a}x.$$

(ii) Sums of exponents.

$$x^{a+b}=x^{a}x^{b}.$$

(iii) Products of exponents.

$$(x^{a})^{b}=x^{ab}.$$

(iv) Powers of the identity.

$$e^{a}=e.$$

(v) Commuting elements. If $xy=yx$, then

$$(xy)^{a}=x^{a}y^{a}.$$

(vi) Powers of units. If $x$ is a unit, with inverse $x^{-1}$ unique by Units of a Monoid: Unique Inverses, Products, and the Group of Units (i), then $x^{a}$ is a unit and

$$(x^{a})^{-1}=(x^{-1})^{a}.$$