I would really appreciate your help with the following question,
Let us define a Preorder T over ${\mathbb R^\mathbb{R}}$ :
- ${ f T g \iff \exists h \in \mathbb R^\mathbb{R}(h \circ g = f)}$
Let ${E = T \cap T^{-1}}$ be an equivalence relation ,
we will define a new relation $T'$ over ${ \mathbb R^\mathbb{R}/E}$ :
- ${[f]_E T' [g]_E\iff fTg }$
we will assume that $T'$ is a partially ordered set and that it is well defined,
prove that there exist Minimum and Maximum in ${<\mathbb R^\mathbb{R}, T'>}$
I started with induction and proved that $[m]_E$ is Min in ${<\mathbb R^\mathbb{R}}$ \ ${\{[a]_E\}, T''>}$
${T'' = \{<x,y> \in T'| x,y \in \mathbb R^\mathbb{R}}$ \ ${\{[a]_E}\}\}$
But got stuck proving that ${<[m]_E,[a]_E>\in T'}$
Thanks in advance!