A new product of algebras and a type reduction theorem (Q795856)

For every finite similarity type \(T\) there is constructed an isomorphism \(F\) of the category of \(T\)-algebras with a full subcategory of the category of groupoids with the following properties: (1) \(F(A)\) is finite iff \(A\) is finite; (2) \(F(A)\) is finitely based iff \(A\) is finitely based; (3) \(\mathrm{End}(F(A))\simeq \mathrm{End}(A)\); \(\mathrm{Sub}(F(A))\simeq \mathrm{Sub}(A)\); \(\mathrm{Con}(F(A))\simeq 1\oplus \mathrm{Con}(A)\). A new product operation, applicable to two algebras of different similarity types, is introduced, used in the construction of \(F\) and also studied separately.