  A fundamental construction of Poisson algebras is as the quasiclassical
limits (QCLs) of associative algebra deformations of commutative associative
algebras. This paper extends this construction to the relative context with the
notion of (bi)module algebras over another algebra for a given algebraic
structure. In this language, a module Poisson algebras can be realized as the
QCLs of bimodule associative deformations of module commutative associative
algebras. Moreover, the notion of the scalar deformation of an $\mathcal
O$-operator is introduced so that the process of bimodule algebras deformations
to QCLs is endowed with $\mathcal O$-operators in a consistent manner. As an
explicit illustration of this process, post-Poisson algebras are realized as
the QCLs of bimodule associative deformations of module commutative associative
algebras with the identity maps as $\mathcal O$-operators, recovering the known
fact that post-Poisson algebras are the QCLs of tridendriform algebra
deformations of commutative tridendriform algebras. Furthermore,
  the notion of scalar deformations of solutions of the associative Yang-Baxter
equation (AYBE) is applied to realize solutions of the Poisson Yang-Baxter
equation (PYBE) in Poisson algebras as solutions of the AYBE in commutative
associative algebras, giving a YBE version of Poisson algebras as the QCLs of
associative deformations of the commutative associative algebras. Finally,
concrete solutions of the PYBE are obtained from the aforementioned
tridendriform deformation-to-QCLs process.
