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We define rank-two and higher rank words and symbolic shifts, extending thenotion of rank-one symbolic shifts of Gao and Hill. We construct infinite wordsof each finite rank, of infinite rank, and show there exist unranked words. Weconsider words generated by substitution schemes and give explicit conditionsfor the word to be at most rank-two and not rank-one. We also study shiftsystems associated with infinite words, including those associated to Sturmiansequences, which we show are rank-two.

On finite symbolic rank words and subshifts