Interpolation and Definability in Extensions of the Minimal Logic

We study into the interpolation property and the projective Beth property in extensions of Johansson's minimal logic. A family of logics of some special form is considered. Effective criteria are specified which allow us to verify whether an arbitrary logic in this family has a given property. Supported by RFBR grant No. 03-06-80178, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1, and by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 726–750, November–December, 2005.     

Minimal Axiomatization in Modal Logic

We consider the problem of finding, in the ambit of modal logic, a minimal characterization for finite Kripke frames, i.e., a formula which, given a frame, axiomatizes its theory employing the lowest possible number of variables and implies the other axiomatizations. We show that every finite transitive frame admits a minimal characterization over K4, and that this result can not be extended to K.     

Exhibiting Wide Families of Maximal Intermediate Propositional Logics with the Disjunction Property

We provide results allowing to state, by the simple inspection of suitable classes of posets (propositional Kripke frames), that the corresponding intermediate propositional logics are maximal among the ones which satisfy the disjunction property. Starting from these results, we directly exhibit, without using the axiom of choice, the Kripke frames semantics of 2^No maximal intermediate propositional logics with the disjunction property. This improves previous evaluations, giving rise to the same conclusion but made with an essential use of the axiom of choice, of the cardinality of the set of the maximal intermediate propositional logics with the disjunction property. Mathematics Subject Classification: 03B55, 03C90.     

All Finitely Axiomatizable Normal Extensions of K4.3 are Decidable

We use the apparatus of the canonical formulas introduced by Zakharyaschev [10] to prove that all finitely axiomatizable normal modal logics containing K4.3 are decidable, though possibly not characterized by classes of finite frames. Our method is purely frame-theoretic. Roughly, given a normal logic L above K4.3, we enumerate effectively a class of (possibly infinite) frames with respect to which L is complete, show how to check effectively whether a frame in the class validates a given formula, and then apply a Harropstyle argument to establish the decidability of L, provided of course that it has finitely many axioms.     

A method of proving interpolation in paraconsistent extensions of the minimal logic

The interpolation property in extensions of Johansson’s minimal logic is investigated. The construction of a matched product of models is proposed, which allows us to prove the interpolation property in a number of known extensions of the minimal logic. It is shown that, unlike superintuitionistic, positive, and negative logics, a sum of J-logics with the interpolation property CIP may fail to possess _CIP, nor even the restricted interpolation property. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 627–648, September–October, 2007.     

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.     

Isotropic Hypersurfaces and Minimal Extensions of Lipschitz Functions

The existence and uniqueness theorem for isotropic hypersurfaces with prescribed boundary in Lorentzian warped products is proved.The proof is based on minimal Lipschitz extensions of functions. __________ Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 39, No. 3, pp. 28–36, 2005 Original Russian Text Copyright #x00A9; by A. A. Klyachin and V. M. Miklyukov     

Definability in Normal Extensions of S4

A projective Beth property, PB2, in normal modal logics extending S4 is studied. A convenient criterion is furnished for PB2 to be valid in a larger family of extensions of K4. All locally tabular extensions of the Grzegorczyk logic with PB2 are described. Superintuitionistic logics with the projective Beth property that have no modal companions with this property are found.     

Semi-Invertible Extensions and Asymptotic Homomorphisms

We consider the semi-group Ext(A, B) of extensions of a separable C ^*-algebra A by a stable C ^*-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semi-group contains the group Ext^–1/2(A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext^–1/2(A, B) is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from C A to M(B) that map S A C( ) A into B.     

Fuzzy logics with modalities

We explore the basic fuzzy logic BL as well as propositional fuzzy logics with modalities □ and ◊ and a total accessibility relation. Formulations and proofs are given to replacement theorems for BL. A basic calculus of modal fuzzy logic is introduced. For this calculus and its extensions, we prove replacement and deduction theorems. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 731–757, November–December, 2006.     

Maximum Monoreflections and Essential Extensions

An extension operator c in a category is an assignment, to each object A a monomorphism c _A : AcA. Seeking to approximate such a c by a functor, in our earlier paper Maximum monoreflections, we showed that with some hypotheses on the category, and on c, there is a monoreflection (c) maximum beneath c. Thus, in a suitable category of rings, using the complete ring of quotients operator Q, each object A has a maximum functorial ring of quotients (Q)A. But the proof gave no hint of how to calculate the general (c)A's, nor the particular (Q)A's. In the present paper, we give an explicit formula (and separate proof of existence) for the (c)A's, under more complicated hypotheses on the category and assuming the c _A 's are essential monomorphisms. We discuss briefly how the formula proves adequate to calculate the (Q)A's in Archimedean f-rings, and some related and necessary constructs in Archimedean l-groups.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

Translation Invariant Asymptotic Homomorphisms and Extensions of <Emphasis Type="Italic">C</Emphasis>*-Algebras

Let A and B be C*-algebras, let A be separable, and let B be σ-unital and stable. We introduce the notion of translation invariance for asymptotic homomorphisms from S A = C₀(?) ? A to B and show that the Connes—Higson construction applied to any extension of A by B is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of A by B from a translation invariant asymptotic homomorphism. This leads to our main result that the homotopy classes of extensions coincide with the homotopy classes of translation invariant asymptotic homomorphisms.     

Modal Aggregation and the Theory of Paraconsistent Filters

This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k-filter of propositions. A k-filter on a non-empty set I is a collection of subsets of I which (i) contains I, (ii) is closed under supersets on I, and (iii) contains ∪{X_i ≤ X_j : 0 ≤ i < j ≤ k} whenever it contains the subsets X₀,…, X_k. The mathematical content of the proof that weakly aggregative modal logic is complete relative to k-ary frame theory, the standard semantic idiom for weakly aggregative modal logic (see [1]) is presented in language-independent terms as a representation theorem for k-filters: every non-trivial k-filter is included in the union of ≤ k non-trivial filters. The elementary theory of k-filters is developed and then applied in the form of an ultrafilter extension result for k-ary frame theory. Mathematics Subject Classification: 03B45.     

基于最小路径与最小割集的复杂系统可靠性的描述与计算

本文讨论基于最小路径和最小割集的复杂系统可靠性的描述与计算问题。引入最小路径矩阵与最小割集矩阵的概念,定义向量间的几种运算,并利用所定义运算给出由子系统可靠度精确表示系统可靠度的解析表达及计算方法。该解析表达非常重要,是复杂系统可靠性理论研究与实际应用的有效工具。     

Higher Degrees of Distributivity in MV-Algebras

In this paper we deal with the (, )-distributivity of an MV-algebra , where and are nonzero cardinals. It is proved that if is singular and (, 2)-distributive, then it is (, )-distributive. We show that if is complete then it can be represented as a direct product of MV-algebras which are homogeneous with respect to higher degrees of distributivity.     

Restricted Interpolation and the Projective Beth Property in Equational Logic

Interconnections between syntactic and categorical properties of equational theories are established. The notions of restricted interpolation and of restricted amalgamation are introduced and their equivalence proved; interrelations of the above-mentioned properties and the projective Beth property, interpolation, and amalgamation are studied.     

On Strict Extensions of Nearness Spaces

Strict extensions of nearness spaces are constructed as spaces of round Cauchy filters. Morita's simple extension is identified as the strict extension generated by Morita-generated filters. Carlson's B-completeness is compared with Herrlich completeness and completeness of Morita T-uniformities. The three completeness concepts are shown to be equivalent in regular nearness spaces.     

Idempotency of Extensions via the Bicompletion

Let Top ₀ be the category of topological T ₀-spaces, QU ₀ the category of quasi-uniform T ₀-spaces, T : QU ₀ → Top ₀ the usual forgetful functor and K : QU ₀ → QU ₀ the bicompletion reflector with unit k : 1 → K. Any T-section F : Top ₀ → QU ₀ is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU ₀ is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005).     

Equivalence of the C*-Algebras qℂ and C 0(ℝ2) in the Asymptotic Category

The results of Kasparov, Connes, Higson, and Loring imply the coincidence of the functors [[qℂ ⊗ K, B ⊗ K]] = [[C ₀(ℝ²) ⊗ K, B ⊗ K]] for any C*-algebra B; here[[A, B]] denotes the set of homotopy classes of asymptotic homomorphisms from A to B. Inthe paper, this assertion is strengthened; namely, it is shown that the algebras qℂ ⊗ K and C ₀(ℝ²) ⊗ K are equivalent in the category whose objects are C*-algebras and morphisms are classes of homotopic asymptotic homomorphisms. Some geometric properties of the obtained equivalence are studied. Namely, the algebras qℂ ⊗ K and C ₀(ℝ²) ⊗ K are represented as fields of C*-algebras; it is proved that the equivalence is not fiber-preserving, i.e., is does not take fibers to fibers. It is also proved that the algebras under consideration are not homotopy equivalent.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 788–796.Original Russian Text Copyright ©2005 by T. V. Shul’man.