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. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, 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 twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...

We prove in this note that, in a variety V, residual finiteness of a finitely presented algebra A is equivalent to the property that any finite partial algebra contained in A is embeddable in a finite F-algebra and each implies that A has a solvable word problem. Finite embeddability. An algebra A is residually finite if for any x # y in A, there is a homomorphism a of A onto a finite algebra such that xct # yu. For the notion of an incomplete or partial algebra in a variety, we refer to [4, 6]. We say that an algebra A in a variety V has the finite embeddability property if any finite incomplete F-algebra contained in A is embeddable in a finite F-algebra. A variety V is said to have the finite embeddability property if every algebra in V has the property. Thus, a variety V has the finite embeddability property if any finite incomplete K-algebra which is embeddable is embeddable in a finite F-algebra. We note also that a variety has the finite embeddability property if its finitely generated algebras have this property. To see this, let A be an algebra in a variety V whose finitely generated algebras have the finite embeddability property and let / be a finite incomplete algebra