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Step by Step  Building Representations in Algebraic Logic
 Journal of Symbolic Logic
, 1995
"... 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 defini ..."
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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...
Synthesising Axioms By Games
"... Contents 1 Introduction 2 2 Story 2 3 To the games 4 4 Generalisations 6 5 Discussion 9 Research partially supported by UK EPSRC grants GR/K54946, GR/L85978, GR/L85961. Thanks to ' Agnes Kurucz and Szabolcs Mikul'as for comments. 1 1 Introduction We would like to begin by hoping that y ..."
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Contents 1 Introduction 2 2 Story 2 3 To the games 4 4 Generalisations 6 5 Discussion 9 Research partially supported by UK EPSRC grants GR/K54946, GR/L85978, GR/L85961. Thanks to ' Agnes Kurucz and Szabolcs Mikul'as for comments. 1 1 Introduction We would like to begin by hoping that you had a very happy birthday (we assume you found more enjoyable ways to spend it than reading this), and we wish you many more years of fruitful research. In this short article we would like to discuss from our current perspective the problem of providing axioms for classes of algebras, and the way in which games can contribute to solving it. Towards the end, we will describe general settings in which this can be done. We will be rather discursive and opinionated  you may well disagree with what we say (and we'd like to hear from you about it), but we are certainly not trying to provoke you or implying that you do disagree with us. We will use natural langua