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Commutative algebras and representations of the category of finite sets
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 21, Tome 388 (2011), pp. 189-195

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We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps. 
@article{ZNSL_2011_388_a7,
     author = {S. S. Podkorytov},
     title = {Commutative algebras and representations of the category of finite sets},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {189--195},
     publisher = {mathdoc},
     volume = {388},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_388_a7/}
}
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S. S. Podkorytov. Commutative algebras and representations of the category of finite sets. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 21, Tome 388 (2011), pp. 189-195. http://geodesic.mathdoc.fr/item/ZNSL_2011_388_a7/