A note on the relation between categories and hyperstructures

In this paper, first we introduce the categories of “n-ary hyperstructures” ({{\mathrm{HS}}_{n}}) and “n-ary hyperstructures with particular carriers and relations” ({{\mathrm{HSP}}_{n}}), and we prove that the categories of{{\mathrm{HS}}_{n}}and{{\mathrm{HSP}}_{n-1}}are isomorphic. Then we prove that the category of “{(n-1)}-ary hypergroupoids” ({{\mathrm{HG}}_{n-1}}) is a full subcategory of the category{{\mathrm{HS}}_{n}}and the category of “unary bindingn-ary algebra with particular properties” ({{\mathrm{UBA}}_{n}}) is a full subcategory of the category{{\mathrm{HSP}}_{n}}. Next, we define two functorsFandG, and show that the restriction of the functorFto the category{{\mathrm{HG}}_{n}}is an isomorphism of{{\mathrm{HG}}_{n}}onto{{\mathrm{UBA}}_{n}}, and the restriction of the functorGto the category{{\mathrm{UBA}}_{n}}is an isomorphism of{{\mathrm{UBA}}_{n}}onto{{\mathrm{HG}}_{n}}.