A generalization of the Mermin and Wagner’s theorem  in a one-dimensional chain with short-range interactions containing both bilinear and biquadratic exchange  is presented. A magnetization per site q m is defined in terms of a statistical average over the Boltzmann distribution of the spin. Using the two point correlation function S yy (k + q) a double commutator and the Bogoliubov inequality, an upper bound is obtained taking into account the result of the double commutator in the presence of the biquadratic exchange term. Following the same steps as for the Mermin and Wagner proof, where only a bilinear exchange term is considered, it is shown that the magnetization, at any finite temperature T, at any value of the coupling constants and for any eigenvalue S, must vanish with the ordering field. Hence, it is not possible in a onedimensional chain with both bilinear and biquadratic exchange to have spontaneous order. Using the obtained double commutator D(k ) and the upper bound it is straightforward to prove the Goldstone theorem in the presence of the biquadratic exchange term under the condition that the equal time correlation function S_zz (k ) diverges.  N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133  R. Zivieri in preparation--Presentazione poster by R. Zivieri-conferenza internazionale
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