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A Survey of Residuated Lattices
"... Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of latticeordered ..."
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Cited by 64 (6 self)
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Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of latticeordered groups, ideal lattices of rings, linear logic and multivalued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability. We end with a list of open problems that we hope will stimulate further research.
PartialGaggles Applied to Logics with Restricted Structural Rules
 In Peter SchroederHeister and Kosta Dosen, editors, Substructural Logics
, 1991
"... Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in ..."
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Cited by 52 (2 self)
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Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in some position. (6) The family of operations OP is founded when there is a distinguished operator f 2 OP (the head) such that any other operator g 2 OP is a relative of f . Definition. A partialgaggle is a tonoid T = (X; ; OP), in which OP is a founded family. As examples, consider a p.o. residuated groupoid, with OP chosen to be any of the following families of operations (ffi is the head of the families of which it is a member): fffig, fffi; /g, fffi; !g, fffi; /;!g, f/g, f!g. Note that f!;/g does not formally fall under our definition since the trace of one is not directly the contrapositive of the trace of the other, even though the trace of each is a contrapositive of the trace of f...
Substructural Logics on Display
, 1998
"... Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek ca ..."
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Cited by 50 (16 self)
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Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponentialfree linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic BiLambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bilinear, Birelevant, BiBCK and Biintuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and som...
Substructural Logics and Residuated Lattices  An Introduction
, 2003
"... This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. Then, residuated lattices are introduced as algebra ..."
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Cited by 32 (3 self)
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This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. Then, residuated lattices are introduced as algebraic structures for substructural logics,
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
Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL
, 2006
"... Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem ..."
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Cited by 20 (6 self)
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Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.
Relevance logic and concurrent composition
 In Proceedings of Third Annual Symposium on Logic in Computer Science
, 1988
"... We show that the operation of relativising properties with respect to parallel environments often employed in obtaining compositionality in theories for concurrency corresponds to a notion of (contraction—free) relevant deduction. We propose to consider program logics in which this notion of deducti ..."
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Cited by 17 (1 self)
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We show that the operation of relativising properties with respect to parallel environments often employed in obtaining compositionality in theories for concurrency corresponds to a notion of (contraction—free) relevant deduction. We propose to consider program logics in which this notion of deduction is internalized by means of the corresponding implication. The idea is carried through for safety properties of a simple system of SCCStype synchronuous processes with an internal choice operator. We present two completeness results; first for a modal extension of positive propositional linear logic w.r.t. the equational class of algebras containing the safety testing quotient of our process system as its free member, and secondly for the free algebra itself.
Algebraic aspects of cut elimination
, 2001
"... We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with ..."
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Cited by 11 (3 self)
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We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasicompletion of these Gentzen structures. It is shown that the quasicompletion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and OkadaTerui [17].