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Hello,

let's suppose we are given a set [itex]A[/itex], a (semi)group [itex]S[/itex] and we define a (semi)group-

Now, if I define a bijection [itex]f:A \rightarrow B[/itex], is it possible to show that there always exists some other (semi)group S' and some action [tex]t':B \times S' \rightarrow B[/tex] such that:

[tex]\forall a \in A[/tex] and [tex]\forall s \in S[/tex]

[tex]f(t(a,s))=t'(f(a),s')[/tex]

for some [tex]s' \in S'[/tex]

let's suppose we are given a set [itex]A[/itex], a (semi)group [itex]S[/itex] and we define a (semi)group-

*action*[itex]t:A \times S \rightarrow A[/itex].Now, if I define a bijection [itex]f:A \rightarrow B[/itex], is it possible to show that there always exists some other (semi)group S' and some action [tex]t':B \times S' \rightarrow B[/tex] such that:

[tex]\forall a \in A[/tex] and [tex]\forall s \in S[/tex]

[tex]f(t(a,s))=t'(f(a),s')[/tex]

for some [tex]s' \in S'[/tex]

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