  In this paper, we define and study (co)homology theories of a compatible
associative algebra $A$. At first, we construct a new graded Lie algebra whose
Maurer-Cartan elements are given by compatible associative structures. Then we
define the cohomology of a compatible associative algebra $A$ and as
applications, we study extensions, deformations and extensibility of finite
order deformations of $A$. We end this paper by considering compatible
presimplicial vector spaces and the homology of compatible associative
algebras.
