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15
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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Cited by 28 (15 self)
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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...
Finite Algebras of Relations Are Representable on Finite Sets
, 1996
"... Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in eac ..."
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Cited by 19 (2 self)
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Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set. 1 Introduction In this paper, we give a simple proof that certain classes K of algebras have the `finite base property'. This will imply decidability of the universal theory of K, and that any finite algebra in K is representable on a finite set. Examples of such K include the relativized cylindric set algebras in dimension n (Crs n ), polyadic Crs, and the weakly associative relation algebras WA. Most of these results were first established in the paper [ABN2]; the original proofs were substantially longer than the present one. What is the finite base property? Say that we are given a class K of concrete algebras. This is to say that the algebras i...
Complete Representations in Algebraic Logic
 JOURNAL OF SYMBOLIC LOGIC
"... A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary. ..."
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Cited by 19 (8 self)
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A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.
Erdös Graphs Resolve Fine's Canonicity Problem
 The Bulletin of Symbolic Logic
, 2003
"... We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any firstorder definable class of relational structures. Using a variant of this construction, we resolve a longstanding question of Fine, by exhibiting a b ..."
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Cited by 11 (8 self)
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We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any firstorder definable class of relational structures. Using a variant of this construction, we resolve a longstanding question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any firstorder definable class of Kripke frames. The constructions use the result of Erd os that there are finite graphs with arbitrarily large chromatic number and girth.
A Note on Relativised Products of Modal Logics
 Advances in Modal Logic
, 2003
"... this paper. each frame of the class.) For example, K is the logic of all nary product frames. It is not hard to see that S5 is the logic of all nary products of universal frames having the same worlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kind as cu ..."
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Cited by 10 (6 self)
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this paper. each frame of the class.) For example, K is the logic of all nary product frames. It is not hard to see that S5 is the logic of all nary products of universal frames having the same worlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kind as cubic universal product S5 frames. Note that the `ireduct' F U 1 U n ; R i of F 1 F n is a union of n disjoint copies of F i . Thus, F and F i validate the same formulas, and so L n L 1 L n : There is a strong interaction between the modal operators of product logics. Every nary product frame satis es the following two properties, for each pair i 6= j, i; j = 1; : : : ; n: Commutativity : 8x8y8z xR i y ^ yR j z ! 9u (xR j u ^ uR i z) ^ xR j y ^ yR i z ! 9u (xR i u ^ uR j z) Church{Rosser property : 8x8y8z xR i y ^ xR j z ! 9u (yR j u ^ zR i u) This means that the corresponding modal interaction formulas 2 i 2 j p $ 2 j 2 i p and 3 i 2 j p ! 2 j 3 i p belong to every ndimensional product logic. The geometrically intuitive manydimensional structure of product frames makes them a perfect tool for constructing formalisms suitable for, say, spatiotemporal representation and reasoning (see e.g. [33, 34]) or reasoning about the behaviour of multiagent systems (see e.g. [4]). However, the price we have to pay for the use of products is an extremely high computational complexityeven the product of two NPcomplete logics can be nonrecursively enumerable (see e.g. [29, 27]). In higher dimensions practically all products of `standard' modal logics are undecidable and non nitely axiomatisable [16]
Nonmodularity Results for Lambda Calculus
 Fundamenta Informaticae
, 2001
"... The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the firstorder predicate logic. In this paper we prove that the lattice of lambda theories is not modular and that the va ..."
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Cited by 10 (7 self)
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The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the firstorder predicate logic. In this paper we prove that the lattice of lambda theories is not modular and that the variety generated by the term algebra of a semisensible lambda theory is not congruence modular. Another result of the paper is that the Mal'cev condition for congruence modularity is inconsistent with the lambda theory generated by equating all the unsolvable lambdaterms.
On the Search for a Finitizable Algebraization of First Order Logic
, 2000
"... We give an algebraic version of first order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of thi ..."
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Cited by 9 (1 self)
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We give an algebraic version of first order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1.1 (a concrete version of which is given by Theorems 2.8 and 4.2), while its logical form is Corollary 5.2. For first order logic with equality we give a result weaker than the one for rst order logic without equality. Namely, in this case  instead of finitely axiomatizing the corresponding class of all representable algebras  we finitely axiomatize only the equational theory of that class. See Subsection 6.1, especially Remark 6.6 there. The proof of Theorem 1.1 is elaborated in Sections 3 and 4. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 4.2 is a purely semigroup theoretic result. Cf. also "Further main results" in the
Relativized Relation Algebras
 Journal of Symbolic Logic
, 1999
"... Relativization is one of the central topics in the study of algebras of relations (i.e. relation and cylindric algebras). Relativized representable relation algebras behave much nicer than the original class RRA: for instance, one obtains finite axiomatizability, decidability and amalgamation by rel ..."
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Cited by 6 (2 self)
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Relativization is one of the central topics in the study of algebras of relations (i.e. relation and cylindric algebras). Relativized representable relation algebras behave much nicer than the original class RRA: for instance, one obtains finite axiomatizability, decidability and amalgamation by relativization. The properties of the class obtained by relativizing RRA depend on the kind of element with which is relativized. We give a systematic account of all interesting choices of relativizing RRA, and show that relativizing with transitive elements forms the borderline where all above mentioned three properties switch from negative to positive. In algebraic logic, relativized cylindric and relation algebras have been studied relatively deeply (cf. e.g., Henkin et al. (Henkin et al., 1981), Maddux (Maddux, 1982) and Resek Thompson (Resek and Thompson, 1991)). The emphasis, however was different from the perspective we will take here. As the name "relativization" indicates, the non...
Strongly representable atom structures of cylindric algebras
, 2007
"... A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n ≥ ..."
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Cited by 5 (1 self)
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A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n ≥ 3, the class of all strongly representable ndimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary. Our proof is based on the following construction. From an arbitrary undirected, loopfree graph Γ, we construct an ndimensional atom structure E(Γ), and prove, for infinite Γ, that E(Γ) is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdős shows that there are graphs Γk (k < ω) with infinite chromatic number, but having a nonprincipal ultraproduct � D Γk whose chromatic number is just two. It follows that E(Γk) is strongly representable (each k < ω) but � D E(Γk) is not. 1
Algebraic Characterizations of two Beth Definability Properties
 Hoogland
, 1996
"... In the present paper we will show that the Beth definability property corresponds to surjectiveness of epimorphisms in abstract algebraic logic. Furthermore, we will give an equally general characterization of the weak Beth definability property, hereby giving a solution to Problem 14 in [Sa 88]. Fi ..."
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Cited by 4 (1 self)
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In the present paper we will show that the Beth definability property corresponds to surjectiveness of epimorphisms in abstract algebraic logic. Furthermore, we will give an equally general characterization of the weak Beth definability property, hereby giving a solution to Problem 14 in [Sa 88]. Finally, we will present two counterexamples showing that two of the more striking assumptions imposed on the logics in our characterizations are necessary. Supported by a grant from the NWO. 1 Contents 1 Introduction 2 2 Preliminaries 6 2.1 Definitions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 Preliminary Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 3 An Algebraic Characterization of the (strong) Beth Definability Property 16 4 An Algebraic Characterization of the Weak Beth Definability Property 23 5 Discussion 28 A Appendix 36 1 Introduction Abstract Algebraic Logic The aim of this paper is to give an algebraic chara...