Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra

One important example of a transposed Poisson algebra can be constructed by means of a commutative algebra and its derivation. This approach can be extended to superalgebras; that is, one can construct a transposed Poisson superalgebra given a commutative superalgebra and its even derivation. In this paper, we show that including odd derivations in the framework of this approach requires introducing a new notion. It is a super vector space with two operations that satisfy the compatibility condition of a transposed Poisson superalgebra. The first operation is determined by a left supermodule over a commutative superalgebra and the second is a Jordan bracket. Then it is proved that the super vector space generated by an odd derivation of a commutative superalgebra satisfies all the requirements of the introduced notion. We also show how to construct a 3-Lie superalgebra if we are given a transposed Poisson superalgebra and its even derivation.