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Let (G, E) be a quasi-ordered group such that E∩E-1 is infinite, (G,G+) an ordered group with G+( )E( )G, and (G, G1) the partially ordered group induced by (G,E).Let γE,G+:( )G+→( )E and γE, G1:( )G1→( )E be the corresponding natural morphisms between T