Categorical theories of a function
Matematičeskie zametki, Tome 11 (1972) no. 1, pp. 89-98.

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In this paper all the theories of signature $\langle f^{(1)}\rangle $ that are categorical in a denumerable power are described, as well as all the theories of this signature that are categorical in nondenumerable powers.
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Yu. E. Shishmarev. Categorical theories of a function. Matematičeskie zametki, Tome 11 (1972) no. 1, pp. 89-98. http://geodesic.mathdoc.fr/item/MZM_1972_11_1_a10/