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103
A Primer On Galois Connections
 York Academy of Science
, 1992
"... : We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are wellknown to most mathematicians who deal with order theory; they seem to be less known to to ..."
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Cited by 49 (4 self)
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: We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are wellknown to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear; and thus, the whole situation typically becomes much easier to understand. KEY WORDS: Galois connection, closure operation, interior operation, polarity, axiality CLASSIFICATION: Primary: 06A15, 0601, 06A06 Secondary: 5401, 54B99, 54H99, 68F05 0. INTRODUCTION Mathematicians are familiar with the following situation: there are two "worlds" and t...
Generalized MValgebras,
 J. Algebra
, 2005
"... Abstract We generalize the notion of an MValgebra in the context of residuated lattices to include noncommutative and unbounded structures. We investigate a number of their properties and prove that they can be obtained from latticeordered groups via a truncation construction that generalizes th ..."
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Cited by 27 (13 self)
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Abstract We generalize the notion of an MValgebra in the context of residuated lattices to include noncommutative and unbounded structures. We investigate a number of their properties and prove that they can be obtained from latticeordered groups via a truncation construction that generalizes the ChangMundici Γ functor. This correspondence extends to a categorical equivalence that generalizes the ones established by D. Mundici and A. Dvurečenskij. The decidability of the equational theory of the variety of generalized MValgebras follows from our analysis.
Pretopologies and completeness proofs
 Journal of Symbolic Logic
, 1995
"... Pretopologies were introduced in [S] and there shown to give a complete semantics for a propositional sequent calculus BL here called basic linear logic1, as well as for its extensions by structural rules, ex falso quodlibet or double negation. Immediately after the Logic Colloquium '88, conver ..."
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Cited by 24 (3 self)
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Pretopologies were introduced in [S] and there shown to give a complete semantics for a propositional sequent calculus BL here called basic linear logic1, as well as for its extensions by structural rules, ex falso quodlibet or double negation. Immediately after the Logic Colloquium '88, conversation with Per MartinLof helped me to see how the pretopology semantics should be extended to predicate logic; the result now is a simple and fully constructive completeness proof for rst order BL and virtually all its extensions, including usual, or structured, intuitionistic and classical logic. Such a proof clearly illustrates the fact that stronger settheoretic principles and classical metalogic are necessary only when completeness is sought with respect to a special class of models, such as usual twovalued models. To make the paper selfcontained, I brie
y review in section 1 the denition of pretopologies; section 2 deals with syntax and section 3 with semantics. The completeness proof in section 4, though similar in structure, is sensibly simpler than that in [S], and this is why it is given in detail. In section 5 it is shown how little is needed to obtain completeness for extensions of BL in the same language. Finally, in section 6 connec
Epistemic actions as resources
 Journal of Logic and Computation
, 2007
"... We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. deceit. We give a purely algebraic treatment of the muddy children puzzle, ..."
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Cited by 24 (18 self)
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We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. deceit. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic actions, that is, information exchanges between agents A, B,... ∈ A, are modeled as elements of a quantale, hence conceiving them as resources. Indeed, quantales are to locales what monoidal closed categories are to Cartesian closed categories, respectively providing semantics for Intuitionistic Logic, and for noncommutative Intuitionistic Linear Logic, including Lambek calculus. The quantale (Q, � , •) acts on an underlying Qright module (M, � ) of epistemic propositions and facts. The epistemic content is encoded by appearance maps, one pair f M A: M → M and f Q A: Q → Q of (lax) morphisms for each agent A ∈ A. By adjunction, they give rise to epistemic modalities [12], capturing the agents ’ knowledge on propositions and actions. The module action is epistemic update and gives rise to dynamic modalities [20] — cf. weakest preconditions. This model subsumes the crucial fragment of Baltag, Moss and Solecki’s [6] dynamic epistemic logic, abstracting it in a constructive fashion while introducing resourcesensitive structure on the epistemic actions. Keywords: Multiagent communication, knowledge update, resourcesensitivity, quantale, Galois adjoints, dynamic epistemic logic, sequent calculus, Lambek calculus, Linear Logic.
Local Possibilistic Logic
 Journal of Applied NonClassical Logic
, 1997
"... Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a tnorm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natur ..."
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Cited by 23 (15 self)
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Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a tnorm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natural interpretation of logical connectives as operations on fuzzy sets. Due to the quantal structure of information states, we obtain a system which shares several features with (exponentialfree) intuitionistic linear logic. Soundness and completeness are proved, parametrically on the choice of the tnorm operation.
Concurrent Kleene Algebra
"... A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partiallyordered trace model of program execution semantics and demonstrate its usefu ..."
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Cited by 22 (3 self)
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A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partiallyordered trace model of program execution semantics and demonstrate its usefulness by validating familiar proof rules for sequential programs (Hoare triples) and for concurrent ones (Jones’s rely/guarantee calculus). This involves an algebraic notion of invariants; for these the exchange inequation strengthens to an equational distributivity law. Most of our reasoning has been checked by computer.
Algebra and Sequent Calculus for Epistemic Actions
 ENTCS PROCEEDINGS OF LOGIC AND COMMUNICATION IN MULTIAGENT SYSTEMS (LCMAS) WORKSHOP, ESSLLI 2004
, 2005
"... We introduce an algebraic approach to Dynamic Epistemic Logic. This approach has the advantage that: (i) its semantics is a transparent algebraic object with a minimal set of primitives from which most ingredients of Dynamic Epistemic Logic arise, (ii) it goes with the introduction of nondeterminis ..."
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Cited by 19 (5 self)
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We introduce an algebraic approach to Dynamic Epistemic Logic. This approach has the advantage that: (i) its semantics is a transparent algebraic object with a minimal set of primitives from which most ingredients of Dynamic Epistemic Logic arise, (ii) it goes with the introduction of nondeterminism, (iii) it naturally extends beyond boolean sets of propositions, up to intuitionistic and nondistributive situations, hence allowing to accommodate constructive computational, informationtheoretic as well as nonclassical physical settings, and (iv) introduces a structure on the actions, which now constitute a quantale. We also introduce a corresponding sequent calculus (which extends Lambek calculus), in which propositions, actions as well as agents appear as resources in a resourcesensitive dynamicepistemic logic.
The Reduced Relative Power Operation on Abstract Domains
 Theor. Comput. Sci
, 1999
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On a Duality of Quantales emerging from an Operational Resolution
 INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS
, 1997
"... We introduce the notion of operational resolution, i.e., an isotone map from a powerset to a poset that meets two additional conditions, which generalizes the description of states as the atoms in a property lattice (Piron, 1976 and Aerts, 1982) or as the underlying set of a closure operator (Aerts, ..."
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Cited by 16 (6 self)
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We introduce the notion of operational resolution, i.e., an isotone map from a powerset to a poset that meets two additional conditions, which generalizes the description of states as the atoms in a property lattice (Piron, 1976 and Aerts, 1982) or as the underlying set of a closure operator (Aerts, 1994 and Moore, 1995). We study the structure preservance of the related state transitions and show how the operational resolution constitutes an epimorphism between two unitary quantales.