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Abstract:We define then-syntactic theories as a natural extension of the syntactic theories. An-syntactic theory is an equational theory which admits a finite presentation in which every proof can be performed with at mostnapplications of an axiom at the root,but no finite presentation in which every proof can be performed with at mostn−1 applications of an axiom at the root. Then-syntactic theories inherit the good properties of the syntactic theories for solving the word problem, or matching or unification problems. We show that for any integern≥ 1, there exists an-syntactic theory.

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