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Macneille completions and canonical extensions
 Transactions of the American Mathematical Society
"... Abstract. Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical exten ..."
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Abstract. Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety V is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure. 1.
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]
Simulating Polyadic Modal Logics by Monadic Ones
 Journal of Symbolic Logic
, 2001
"... We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic in the diamond language with each logic in the language with polyadic modal connectives. We prove that t ..."
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Cited by 6 (2 self)
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We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic in the diamond language with each logic in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and firstorder definability.
Open Problems in Logic and Games
 Logical Construction Games', Acta Philosophica Fennica 78, T. Aho & AV Pietarinen, eds., Truth and Games, essays in honour of Gabriel Sandu, 123  138. J. van Benthem, 2006B, 'The Epistemic Logic of IF Games', in
, 2005
"... Dov Gabbay is a prolific logician just by himself. But beyond that, he is quite good at making other people investigate the many further things he cares about. As a result, King's College London has become a powerful attractor in our field worldwide. Thus, it is a great pleasure to be an organizer f ..."
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Cited by 6 (2 self)
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Dov Gabbay is a prolific logician just by himself. But beyond that, he is quite good at making other people investigate the many further things he cares about. As a result, King's College London has become a powerful attractor in our field worldwide. Thus, it is a great pleasure to be an organizer for one of its flagship events: the Augustus de Morgan Workshop of 2005. Benedikt Loewe and I proposed the topic of 'interactive logic ' for this occasion, with an emphasis on social software – the logical analysis and design of social procedures – and on games, arguably the formal interactive setting par excellence. This choice reflects current research interests in our logic community at ILLC Amsterdam and beyond. In this broad area of interfaces between logic, computer science, and game theory, this paper is my own attempt at playing Dov. I am, perhaps not telling, but at least asking other people to find out for me what I myself cannot. A word of historical clarification may help here. The last time the Dutch came up the Thames (in 1667), we messed up the harbour, burnt down a few buildings, and took the English flagship the Royal Charles with us as a souvenir. The Medway Raid was still commemorated as late as 1967 in a joint ceremony. This time, however, our intentions
A comprehensive combination framework
, 2006
"... We define a general notion of a fragment within higherorder type theory; a procedure for constraint satisfiability in combined fragments is outlined, following NelsonOppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitab ..."
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Cited by 6 (3 self)
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We define a general notion of a fragment within higherorder type theory; a procedure for constraint satisfiability in combined fragments is outlined, following NelsonOppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitable noetherianity conditions and admits an abstract version of a ‘KeislerShelah like’ isomorphism theorem. We show that this general decidability transfer result covers recent work on combination in firstorder theories as well as in various intensional logics such as description, modal, and temporal logics.
Logic of spacetime and relativity theory
, 2006
"... 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4 ..."
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Cited by 5 (3 self)
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2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4
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
A note on complex algebras of semigroups
 Relational and KleeneAlgebraic Methods in Computer Science: Proc. 7th Int. Sem. Relational Methods in Computer Science and 2nd Int. Workshop Applications of Kleene Algebra
, 2003
"... Abstract. The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups. An example is given of an 8element commutative B ..."
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Abstract. The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups. An example is given of an 8element commutative Boolean semigroup that is not in this variety, and an analysis of all smaller Boolean semigroups shows that there is no smaller example. However, without associativity the situation is quite different: the variety generated by complex algebras of (commutative) binars is finitely based and is equal to the variety of all Boolean algebras with a (commutative) binary operator. A binar is a set A with a (total) binary operation ·, and in a partial binar this operation is allowed to be partial. We write x · y ∈ A to indicate that the product of x and y exists. A partial semigroup is an associative partial binar, i.e. for all x, y, z ∈ A, if (x · y) · z ∈ A or x · (y · z) ∈ A, then both terms exist and evaluate to the same element of A. Similarly, a commutative partial binar is a binar such that if x · y ∈ A then x · y = y · x ∈ A. Let (P)(C)Bn and (P)(C)Sg denote the class of all (partial) (commutative) groupoids and all (partial) (commutative) semigroups respectively. For A ∈ PBn the complex algebra of A is defined as Cm(A) = 〈P (A), ∪, ∅, ∩, A, \, ·〉, where X · Y = {x · y  x ∈ X, y ∈ Y and x · y exists} is the complex product of X, Y ∈ Cm(A). Algebras of the form Cm(A) are examples of Boolean algebras with a binary operator, i.e., algebras 〈B, ∨, 0, ∧, 1, ¬, ·〉 such that 〈B, ∨, 0, ∧, 1, ¬ 〉 is a Boolean algebra and · is a binary operation that distributes over finite (including empty) joins in each argument. A Boolean semigroup is a Boolean algebra with an associative binary operator. For a class K of algebras, Cm(K) denotes the class of all complex algebras of K, H(K) is the class of all homomorphic images of K, and V(K) is the variety generated by K, i.e., the smallest equationally defined class that contains K. The aim of this note is to contrast the equational theory of Cm((C)Bn) with that of Cm((C)Sg). It turns out that the former is finitely based while the latter is not.