Dilation of generalized Toeplitz kernels on ordered groups

We introduce the concept of Toeplitz-Kre?n-Cotlar triplet on ordered groups and we prove a general dilation result and a general representation result for the positive definite case.This general result includes and extends previous generalizations of the Kre?n extension theorem, the Sz.-Nagy and Foias commutant lifting theorem and the generalized Herglotz-Bochner-Weil theorem.