Berezinians in substitution structures (Q1123454)

This note is an attempt to approach a hypothetical notion of a supermanifold not relying on any conventional function- or sheaf- theoretic foundation. In this connection D. A. Leites is quoted as saying that F. A. Berezin himself insisted that supermanifolds should be defined without a reference to points, and considered the first definition of a supermanifold [\textit{F. A. Berezin} and \textit{D. A. Leites}, Dokl. Akad. Nauk SSSR 224, 505-508 (1975; Zbl 0331.58005)] incomplete. The key notion is that of a substitution algebra, which formalizes the notion of a superfunction algebra over a superdomain, together with its group of coordinate transformations. In such superalgebras certain finite subsets called coordinate systems are distinguished. Some substitutional algebras F admit left partial derivatives by coordinates (elements of coordinate systems); these derivatives are derivations of F of a special type. In this context, a fresh look at the notion of the Berezinian \((=\sup er\det er\min ant)\) of a coordinate change is given.