Homomorphism of Powers/Naturally Ordered Semigroup

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Theorem

Let $\struct {T_1, \odot}$ and $\struct {T_2, \oplus}$ be semigroups.

Let $\phi: \struct {T_1, \odot} \to \struct {T_2, \oplus}$ be a (semigroup) homomorphism.


Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

For a given $a \in T_1$, let $\map {\odot^n} a$ be the $n$th power of $a$ in $T_1$.

For a given $a \in T_2$, let $\map {\oplus^n} a$ be the $n$th power of $a$ in $T_2$.

Then:

$\forall a \in T_1: \forall n \in \struct {S^*, \circ, \preceq}: \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a}$

where $S^* = S \setminus \set 0$.


Proof

The proof proceeds by the Principle of Mathematical Induction for a Naturally Ordered Semigroup.


Let $A := \set {n \in S^*: \forall a \in T_1: \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a} }$

That is, $A$ is defined as the set of all $n$ such that:

$\forall a \in T_1 \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a}$


Basis for the Induction

We have that:

\(\displaystyle \map \phi {\map {\odot^1} a}\) \(=\) \(\displaystyle \map \phi a\) Definition of Power of Element of Magma
\(\displaystyle \) \(=\) \(\displaystyle \map {\oplus^1} {\map \phi a}\) Definition of Power of Element of Magma

So $1 \in A$.

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $k \in A$ where $k \ge 1$, then it logically follows that $k \circ 1 \in A$.


So this is our induction hypothesis:

$\forall a \in T_1: \map \phi {\map {\odot^k} a} = \map {\oplus^k} {\map \phi a}$


Then we need to show:

$\forall a \in T_1: \map \phi {\map {\odot^{k \circ 1} } a} = \map {\oplus^{k \circ 1} } {\map \phi a}$


Induction Step

This is our induction step:

\(\displaystyle \map \phi {\map {\odot^{k \circ 1} } a}\) \(=\) \(\displaystyle \map \phi {\paren {\map {\odot^k} a} \odot a}\) Definition of Power of Element of Magma
\(\displaystyle \) \(=\) \(\displaystyle \paren {\map \phi {\map {\odot^k} a} } \oplus \paren {\map \phi a}\) Definition of Semigroup Homomorphism
\(\displaystyle \) \(=\) \(\displaystyle \paren {\map {\oplus^k} {\map \phi a} } \oplus \paren {\map \phi a}\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \map {\oplus^{k \circ 1} } {\map \phi a}\) Definition of Power of Element of Magma

So $k \in A \implies k \circ 1 \in A$ and the result follows by the Principle of Mathematical Induction:

$\forall n \in \struct {S^*, \circ, \preceq}: \map \phi {\map {\odot^n} a} = \map {\oplus^n} {\map \phi a}$

$\blacksquare$


Sources