Theories, Pre-Theories and Finite State Transformations on Trees
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The closure of an algebra is defined as a generalization of the semigroup of a finite automation. Pretheories are defined as a subclass of the closed algebras, and the relationship between pretheories and the algebraic theories of Lawrence  is explored. Finally, pretheories are applied to the characterization problem of finite state transformations on trees, solving an open problem of Thatcher .