  Interrelations between discrete deformations of the structure constants for
associative algebras and discrete integrable systems are reviewed. A theory of
deformations for associative algebras is presented. Closed left ideal generated
by the elements representing the multiplication table plays a central role in
this theory. Deformations of the structure constants are generated by the
Deformation Driving Algebra and governed by the central system of equations. It
is demonstrated that many discrete equations like discrete Boussinesq equation,
discrete WDVV equation, discrete Schwarzian KP and BKP equations, discrete
Hirota-Miwa equations for KP and BKP hierarchies are particular realizations of
the central system. An interaction between the theories of discrete integrable
systems and discrete deformations of associative algebras is reciprocal and
fruitful.An interpretation of the Menelaus relation (discrete Schwarzian KP
equation), discrete Hirota-Miwa equation for KP hierarchy, consistency around
the cube as the associativity conditions and the concept of gauge equivalence,
for instance, between the Menelaus and KP configurations are particular
examples.
