A computable isomorphism is an isomorphism between presented structures whose underlying map is a total computable function. For presentations on , its inverse is computable too: enumerate inputs until the required output occurs. Two abstractly isomorphic structures with a recursive presentation of a structure need not admit a computable isomorphism between the chosen presentations.
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Computable isomorphism, in the context of mathematical logic and computability theory, refers to a specific type of isomorphism between two structures (usually algebraic structures like groups, rings, etc.) that can be effectively computed by a Turing machine.