# Green's relations

1. Feb 13, 2013

### Ted123

1. The problem statement, all variables and given/known data

If $\phi : S \to T$ is a semigroup morphism then show that if $a\; \mathcal{R} \;b$ in $S$ then $a\phi \; \mathcal{R} \; b\phi$ in $T$.

2. Relevant equations

Recall that if $S$ is a semigroup then for $a\in S$ $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$ The relation $\mathcal{R}$ on a semigroup $S$ is defined by the rule: $$a\;\mathcal{R}\; b \Leftrightarrow aS^1 = bS^1 \;\;\;\;\forall \;\;a,b\in S\text{.}$$

If $S,T$ are semigroups and $\phi : S \to T$ is a (homo)morphism then we apply the map $\phi$ from the right and write: $a\phi = \phi(a)$.

3. The attempt at a solution

$a \; \mathcal{R} \; b \iff aS^1 = bS^1 \iff \exists \; s,t \in S^1$ with $a=bs$ and $b=at \iff a=b$ or $\exists \; s,t \in S$ with $a=bs$ and $b=at$.

How do I get from here to the result? Do $s\phi$ and $t\phi$ satisfy this relation in $T$? If so how would this be written in proofy terms?