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An algebraic representation of the Turing machines is given, where theconfigurations of Turing machines are represented by 4 order tensors, and thetransition functions by 8 order tensors. Two types of tensor product aredefined, one is to model the evolution of the Turing machines, and the other isto model the compositions of transition functions. It is shown that the twotypes of tensor product are harmonic in the sense that the associate law isobeyed.

On the Algebraic Representation of One-Tape Deterministic Turing Machine