Step by Step - Building Representations in Algebraic Logic

We consider the problem of nding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions deening representable relation algebras (for the nite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by rst order formulas. Equivalence games are deened, and are used to establish whether an algebra is !-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all repre-sentable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method.