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Green's relations

  1. Feb 13, 2013 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

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

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

    3. The attempt at a solution

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

    How do I get from here to the result? Do [itex]s\phi[/itex] and [itex]t\phi[/itex] satisfy this relation in [itex]T[/itex]? If so how would this be written in proofy terms?
  2. jcsd
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