Solutions To The Poisson Yang-Baxter Equation And (?)-operators Of Poisson Superalgebras

We investigate connections between even and odd (?)-operators of Poisson superalgebras.We show that a solution to the Poisson Yang-Baxter equation(PYBE)can be interpreted as an (?)-operator associated to the co-regular representation of a Poisson superalgebra.Moreover,an (?)-operator associated to a general representation give a solution to the PYBE in the semi-direct product Poisson superalgebra.We also reveal the relationship between (?)-operators and compatible pre-Poisson superalgebra structures on a Poisson superalgebra.We finally use a coherent pre-Poisson superalgebra to construct both even and odd solutions to the PYBE in(different)semi-direct product Poisson superalgebras.