On the interpolation property of some intuitionistic modal logics

LetL be one of the intuitionistic modal logics considered in [7] (or one of its extensions) and letM _L be the algebraic semantics ofL. In this paper we will extend toL the equivalence, proved in the classical case (see [6]), among he weak Craig interpolation theorem, the Robinson theorem and the amalgamation property of varietyM _L. We will also prove the equivalence between the Craig interpolation theorem and the super-amalgamation property of varietyM _L. Then we obtain the Craig interpolation theorem and Robinson theorem for two intuitionistic modal logics, one ofS ₄-type and the other one ofS ₅-type, showing the super-amalgamation property of the corresponding algebraic semantics.     

On the Finite Model Property of Intuitionistic Modal Logics over MIPC

MIPC is a well-known intuitionistic modal logic of Prior (1957) and Bull (1966). It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property whenever L is Kripke-complete and universal.     

Typed lambda-calculus in classical Zermelo-Frænkel set theory

system of simple types  , which uses the intuitionistic propositional calculus, with the only connective →. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property: every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard [4], under the name of system F, still with the normalization property. More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming languages. It is interesting to notice that, as early as 1972, Clint and Hoare [1] had made an analogous remark for the law of excluded middle and controlled jump instructions in imperative languages. There are now many type systems which are based on classical logic; among the best known are the system LC of J.-Y. Girard [5] and the λμ-calculus of M. Parigot [11]. We shall use below a system closely related to the latter, called the λ _c -calculus [8, 9]. Both systems use classical second order logic and have the normalization property. In the sequel, we shall extend the λ _c -calculus to the Zermelo-Fr?nkel set theory. The main problem is due to the axiom of extensionality. To overcome this difficulty, we first give the axioms of ZF in a suitable (equivalent) form, which we call ZF _ɛ . Received: 6 September 1999 / Published online: 25 January 2001     

Irreducible Quasi-Finite Representations of a Block Type Lie Algebra

Let B be the Block type Lie algebra over ? with basis {L _{α, i} , C ₁, C ₂ | (α, i) ∈ ? × ? \ {(0, ? 2)}} and Lie bracket [L _{α, i} , L _{β, j} ] = (β(i + 1) ? α(j + 1))L _{α+β, i+j}  + αδ_{α+β, 0}δ _{i+j, ?2} C ₁ + (i + 1)δ_{α+β, 0}δ _{i+j, ?2} C ₂, where C ₁, C ₂ are central elements. In this paper, it is proved that a quasi-finite irreducible B-module is either a highest or a lowest weight module. We also give a classification of all highest/lowest weight B-modules.     

A generalization ofL-splines

A theory of generalized splines is developed for all regular formally self adjoint differential operatorsL with real coefficients. A special case of such operators are those which may be factored in the formL =L ₁ ^* L ₁, such as those related to the generalized splines of Ahlberg, Nilson, and Walsh [1, 2], and theL-splines of Schultz and Varga [6]. Theorems giving unique interpolation, integral relations, and convergence rates are established. IfL has a certain positivity property, a useful extremal result is proven.This research was supported in part by a NASA Traineeship, at the Georgia Institute of Technology.     

Quantum symmetric pairs and representations of double affine Hecke algebras of type {C^\vee C_n}

We build representations of the affine and double affine braid groups and Hecke algebras of type C^ú C_n{C^\vee C_n} based upon the theory of quantum symmetric pairs (U, B). In the case U=U_q(\mathfrakgl_N){{\bf U}=\mathcal{U}_{\rm q}(\mathfrak{gl}_N)}, our constructions provide a quantization of the representations constructed by Etingof, Freund, and Ma in [15], and also a type C^ú C_n{C^\vee C_n} generalization of the results in [19].     

Simultaneous reduction of a lattice basis and its reciprocal basis

Given a latticeL we are looking for a basisB=[b ₁, ...b _n ] ofL with the property that bothB and the associated basisB ^*=[b ₁ ^* , ...,b _n ^* ] of the reciprocal latticeL ^* consist of short vectors. For any such basisB with reciprocal basisB ^* let . Håstad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)exp(c ₁·n ^1/3) for a constantc ₁ independent ofn. We improve this upper bound toS(B)exp(c ₂·(lnn)²) withc ₂ independent ofn.We will also introduce some new kinds of lattice basis reduction and an algorithm to compute one of them. The new algorithm proceeds by reducing the quantity . In combination with an exhaustive search procedure, one obtains an algorithm to compute the shortest vector and a Korkine-Zolotarev reduced basis of a lattice that is efficient in practice for dimension up to 30.     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

On normed products of operator ideals which contain $ \frak L_2 $ as a factor

We investigate quasi-Banach operator ideal products (\frak A°\frak B,A°B) (\frak A\circ\frak B,\textbf{A}\circ \textbf{B}) which contain (\frak L₂, L₂) (\frak L_2, \textbf{L}_2) as a factor. In particular, we ask for conditions which guarantee that A°B \textbf{A}\circ \textbf{B} is even a norm if each factor of the product is a 1-Banach ideal. In doing so, we reveal the strong influence of the existence of such a norm in relation to the accessibility of the product ideal and the structure of its factors.     

Sequential Reflexive Logics with Noncontingency Operator

Hilbert systems L and sequential calculi [L ] for the versions of logics L= T,S4,B,S5, and Grz stated in a language with the single modal noncontingency operator A=A¬ A are constructed. It is proved that cut is not eliminable in the calculi [L ], but we can restrict ourselves to analytic cut preserving the subformula property. Thus the calculi [T ], [S4 ], [S5 ] ([Grz ], respectively) satisfy the (weak, respectively) subformula property; for [B ₂ ], this question remains open. For the noncontingency logics in question, the Craig interpolation property is established.     

The relation between intuitionistic and classical modal logics

Intuitionistic propositional logicInt and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded as just fragments of classical modal logics containingS4. The main aim of this paper is to construct a similar correspondence between intermediate logics augmented with modal operators—we call them intuitionistic modal logics—and classical polymodal logics We study the class of intuitionistic polymodal logics in which modal operators satisfy only the congruence rules and so may be treated as various sorts of □ and ◇. Supported by the Alexander von Humboldt Foundation. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 121–155, March–April, 1997.     

A class of symmetric graphs with 2‐arc transitive quotients

Let Γ be an X‐symmetric graph admitting an X‐invariant partition ?? on V(Γ) such that Γ_?? is connected and (X, 2)‐arc transitive. A characterization of (Γ, X, ??) was given in [S. Zhou Eur J Comb 23 (2002), 741–760] for the case where |B|>|Γ(C)∩B|=2 for an arc (B, C) of Γ_??.We con‐sider in this article the case where |B|>|Γ(C)∩B|=3, and prove that Γ can be constructed from a 2‐arc transitive graph of valency 4 or 7 unless its connected components are isomorphic to 3 K ₂, C ₆ or K _{3, 3}. As a byproduct, we prove that each connected tetravalent (X, 2)‐transitive graph is either the complete graph K ₅ or a near n‐gonal graph for some n?4. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 232–245, 2010     

Holomorphic functional calculus of Hodge-Dirac operators in L p

We study the boundedness of the H ^∞ functional calculus for differential operators acting in L ^p (R ⁿ ; C ^N ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the L ^p theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients Π _B as treated in L ²(R ⁿ ; C ^N ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π _B has a bounded H ^∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under small perturbations of this functional calculus.     

Varieties of restriction semigroups and varieties of categories

The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, ^?1) by forgetting the inverse operation and retaining the two operations x⁺ = xx^?1 and x* = x^?1x. The subvariety B of strict restriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B₂, B₂M = B] and [B₀, B₀M]. Here, B₂ and B₀ are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B₂, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson's major theorems have natural interpretations and application to the interval [B₂, B] and, with modification, to the interval [B₀, B₀M] that lies below it. Further exploration may lead to applications in the reverse direction.     

The Yoneda Algebra of a Graded Ore Extension

Let A be a connected-graded algebra with trivial module 𝕜, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A): = Ext _A (𝕜, 𝕜) to E(B). Cassidy and Shelton have shown that when A satisfies their 𝒦₂ property, B will also be 𝒦₂. We prove the converse of this result.     

On the theta completions for maximal subalgebras

Let L be a finite dimensional Lie algebra. Then for a maximal subalgebra M of L, a 𝜃-completion for M is a subalgebra C of L such that CM and M_L?C and C∕M_L contains no non-zero ideal of L∕M_L, properly. And a 𝜃-completion C of M is said to be a strong 𝜃-completion, if C = L or there exists a subalgebra B of L such that C be maximal in B and B is not a 𝜃-completion for M. These are analogous to the concepts of 𝜃-completion and strong 𝜃-completion of a maximal subgroup of a finite group. Now, we consider the influence of these concepts on the structure of a finite dimensional Lie algebra.     

Co-rectangular bands and cosheaves in categories of algebras

Conditions on a categoryC are studied which imply that every structure of rectangular band on an objectS ofC arises from a unique product decompositionS=S ₁×S ₂, especially in the case whereC is the opposite of a category of algebras.Sheaves on Stone spaces with values in opposites of categories of algebras are examined.The analog of the bounded Boolean power constructionR[B]^* forR an object of a general category is described.This work was done while the author was partly supported by NSF contract DMS 85-02330.Presented by R. S. Pierce.     

Analytic completeness theorem for singular biprobability models

The aim of the paper is to prove tha analytic completeness theorem for a logic L(∫₁, ∫₂)_A^s with two integral operators in the singular case. The case of absolute continuity was proved in [4]. MSC: 03B48, 03C70.     

Hypersequent calculi for intuitionistic logic with classical atoms

We discuss a propositional logic which combines classical reasoning with constructive reasoning, i.e., intuitionistic logic augmented with a class of propositional variables for which we postulate the decidability property. We call it intuitionistic logic with classical atoms. We introduce two hypersequent calculi for this logic. Our main results presented here are cut-elimination with the subformula property for the calculi. As corollaries, we show decidability, an extended form of the disjunction property, the existence of embedding into an intuitionistic modal logic and a partial form of interpolation.     

Sémantique algébrique ďun système logique basé sur un ensemble ordonné fini

In order to modelize the reasoning of an intelligent agent represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this papers, we consider logic systems called L′_T without this kind of constants but limited to the case where T is a finite poset. We prove a weak deduction theorem. We introduce also an algebraic semantics using Hey ting algebra with operators. To prove the completeness theorem of the L′_T system with respect to the algebraic semantics, we use the method of H. Rasiowa and R. Sikorski for first order logic. In the propositional case, a corollary allows us to assert that it is decidable to know “if a propositional formula is valid”. We study also certain relations between the L′_T logic and the intuitionistic and classical logics.