We consider the problem of finding 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' defining
representable relation algebras (for the finite case) and a similar schema
for cylindric algebras are derived. Finite relation algebras with homogeneous
representations are characterised by first order formulas. Equivalence
games are defined, and are used to establish whether an algebra is $\omega$-categorical.
We have a simple proof that the perfect extension of a representable relation
algebra is completely representable. Another two-player game is used to
derive equational axiomatisations for the classes of all representable
relation algebras and representable cylindric algebras.
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