Up to equimorphism, hyperarithmetic is recursive

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Abstract
Two linear orderings areequimorphicif each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one.On the way to our main result we prove that a linear ordering has Hausdorff rank less thanif and only if it is equimorphic to a recursive one. As a corollary of our proof we prove that, given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of Hausdorff rank at most α ordered by embeddablity is recursively presentable.