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An algebra isomorphism between algebras of matrices and difference operatorsis used to investigate the discrete integrable hierarchy. We find local andnon-local families of R-matrix solutions to the modified Yang-Baxter equation.The three R-theoretic Poisson structures and the Suris quadratic bracket arederived. The resulting family of bi-Poisson structures include a seminaldiscrete bi-Poisson structure of Kupershmidt at a special value.Comment: 22 pages, LaTeX, v3: Minor change

COMPATIBLE POISSON STRUCTURES OF TODA TYPE DISCRETE HIERARCHY