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14
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...
Relation Algebra Reducts of Cylindric Algebras and an Application to Proof Theory
, 1998
"... We confirm a conjecture about neat embeddings of cylindric algebras made in 1969 by J. D. Monk, confirm a later conjecture by Maddux about relation algebras obtained from cylindric algebras, and solve a problem raised by Tarski and Givant. These results in algebraic logic have the following conseque ..."
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Cited by 9 (3 self)
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We confirm a conjecture about neat embeddings of cylindric algebras made in 1969 by J. D. Monk, confirm a later conjecture by Maddux about relation algebras obtained from cylindric algebras, and solve a problem raised by Tarski and Givant. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal ff 3 there is a logically valid sentence ', in a firstorder language with equality and exactly one nonlogical binary relation symbol, such that ' contains only 3 variables (each of which can occur arbitrarily many times), ' has a proof containing exactly ff+1 variables, but no proof containing only ff variables. 1 Introduction and summary Cylindric algebras are an algebraisation of the theory of ffary relations. For the special case of binary relations, there is an alternative approach to algebraisation, using relation algebras. This paper is concerned with the connection between the two approaches and the consequences for ffvariable p...
Axiomatising Various Classes of Relation and Cylindric Algebras
, 1997
"... We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras. ..."
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Cited by 8 (5 self)
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We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras.
THE HYPERRING OF ADÈLE CLASSES
"... Abstract. We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK = AK/K × of a global field K. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G ⊂ ..."
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Abstract. We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK = AK/K × of a global field K. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G ⊂ R × is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G ∪ {0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semilinear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous nonDesarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the
Provability with Finitely Many Variables
"... For every finite n>=4 there is a logically valid sentence 'n with the following properties: 'n contains only 3 variables (each of which occurs many times); 'n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); &apo ..."
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For every finite n>=4 there is a logically valid sentence 'n with the following properties: 'n contains only 3 variables (each of which occurs many times); 'n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); 'n has a proof in firstorder logic with equality that contains exactly n variables, but no proof containing only n \Gamma 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) firstorder binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that 'n has a proof with only n variables. To show that 'n has no proof with only n \Gamma 1 variables we use alternative semantics in place of the usual, standard, settheoretical semantics of firstorder logic.
Algebraic Formalisations of Fuzzy Relations and Their Representation Theorems
, 1998
"... The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representati ..."
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The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representation theorems. The calculus of relations has been investigated since the middle of the nineteenth century. The modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski. The categorical approach to relational calculus was initiated by Mac Lane and Puppe, and Dedekind categories were introduced by Olivier and Serrato. The representation problem for Boolean relation algebras was proposed by Tarski as the question whether every Boolean relation algebra is isomorphic to an algebra of ordinary homogeneous relations. There are many sufficient conditions that guarantee representability for Boolean relation algebras. Schmidt and Strohlein gave a simple proof of the...
Modal logic of projective geometries of finite dimension
, 1998
"... Extending an original idea of B. Jonsson, R. Lyndon showed in [Lyndon 1961] how to construct relation algebras  boolean algebras with the binary operator � (composition), the unary operator ^ (converse) and the constant 1 ' (identity) which satisfy the socalled Tarski axioms (see [Henkin et a ..."
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Extending an original idea of B. Jonsson, R. Lyndon showed in [Lyndon 1961] how to construct relation algebras  boolean algebras with the binary operator � (composition), the unary operator ^ (converse) and the constant 1 ' (identity) which satisfy the socalled Tarski axioms (see [Henkin et alii 1997], [Jonsson & Tarski 1952])  from projective geometries, thus providing "a
Undecidable theories of Lyndon algebras
 J. Symbolic Logic
, 2001
"... With each projective geometry we can associate a Lyndon algebra. Such an algebra always satisfies Tarski’s axioms for relation algebras and Lyndon algebras thus form an interesting connection between the fields of projective geometry and algebraic logic. In this paper we prove that if G is a class o ..."
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With each projective geometry we can associate a Lyndon algebra. Such an algebra always satisfies Tarski’s axioms for relation algebras and Lyndon algebras thus form an interesting connection between the fields of projective geometry and algebraic logic. In this paper we prove that if G is a class of projective geometries which contains an infinite projective geometry of dimension at least three, then the class L(G) of Lyndon algebras associated with projective geometries in G has an undecidable equational theory. In our proof we develop and use a connection between projective geometries and diagonalfree cylindric algebras. 1
Relation algebras and their application in qualitative spatial reasoning
, 2005
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Atom Conjecture
"... Let KN denote the complete graph on N vertices with vertex set V = V (KN) and edge set E = E(KN). For x, y ∈ V, let xy denote the edge between the two vertices x and y. Let L be any finite set and M ⊆ L 3. Let c: E → L. Let [n] denote the integer set {1, 2,..., n}. For x, y, z ∈ V, let c(xyz) denote ..."
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Let KN denote the complete graph on N vertices with vertex set V = V (KN) and edge set E = E(KN). For x, y ∈ V, let xy denote the edge between the two vertices x and y. Let L be any finite set and M ⊆ L 3. Let c: E → L. Let [n] denote the integer set {1, 2,..., n}. For x, y, z ∈ V, let c(xyz) denote the ordered triple (c(xy), c(yz), c(xz)). We say that c is good with respect to M if the following conditions obtain: (i) ∀x, y ∈ V and ∀(c(xy), j, k) ∈ M, ∃z ∈ V such that c(xyz) = (c(xy), j, k); (ii) ∀x, y, z ∈ V, c(xyz) ∈ M; and (iii) ∀x ∈ V ∀ℓ ∈ L ∃ y ∈ V such that c(xy) = ℓ. We investigate particular subsets M ⊆ L3 and those edge colorings of KN which are good with respect to these subsets M. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture. 1 Motivation and