On the Beth properties of some intuitionistic modal logics

 Let L be one of the intuitionistic modal logics considered in [4]. As in the classical modal case (see [7]), we define two different forms of the Beth property for L, which are denoted by B ₁ and B ₂ ; in this paper we study the relation among B ₁ ,B ₂ and the interpolation properties C ₁ and C ₂ , introduced in [4]. It turns out that C ₁ implies B ₁ , but contrary to the boolean case, is not equivalent to B ₁ . It is shown that B ₂ and C ₂ are independent, and moreover it comes out that, in contrast to classical case, there exists an extension of the intuitionistic modal logic of S ₄ -type, that has not the property B ₂ . Finally we give two algebraic properties, that characterize respectively B ₁ and B ₂ . Received: 27 September 1999 / Published online: 31 May 2002     

The Disjunction Property in the Class of Paraconsistent Extensions of Minimal Logic

We consider the disjunction property, DP, in the class of extensions of minimal logic L _j . Conditions are described under which DP is translated from the class PAR of properly paraconsistent extensions of the logics of class L _j into the class INT of intermediate extensions and the class NEG of negative extensions, and conditions for its being translated back into PAR. The logic L _F in PAR, which specifies conditions for DP to be translated from PAR into NEG, is defined and is characterized in terms of j-algebras and Kripke frames. Moreover, we show that L _F is decidable and possesses the disjunction property.     

Finite-to-finite universal quasivarieties are Q-universal

Let K be a quasivariety of algebraic systems of finite type. K is said to be universal if the category G of all directed graphs is isomorphic to a full subcategory of K. If an embedding of G may be effected by a functor F:G K which assigns a finite algebraic system to each finite graph, then K is said to be finite-to-finite universal. K is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.?We establish a connection between these two, apparently unrelated, notions by showing that if K is finite-to-finite universal, then K is Q-universal. Using this connection a number of quasivarieties are shown to be Q-universal. Received February 8, 2000; accepted in final form December 23, 2000.     

Singular Archimedean lattice-ordered groups

W stands for the category of all archimedean l-groups with designated weak unit. The subcategory W _s of all groups with singular weak unit is analyzed as a full subcategory of W which is both epireflective and monocoreflective. A general technique for "contracting" monoreflections of a category A to a monocoreflective subcategory B is developed and then applied to W _s to show that: (i) the projectable hull in W _s is a monoreflection; (ii) essential hulls in W _s are formed by simply taking the lateral completion, and G is essentially closed in this category if and only if , where X is compact, Hausdorff and extremally disconnected; (iii) the maximum monoreflection on W _s , denoted , is obtained by contracting the maximum monoreflection on W, and G is epicomplete in W _s precisely when G is laterally -complete; (iv) the maximum essential reflection on W _s , denoted , is the contraction of the maximum essential reflection on W. Received January 22, 1997; accepted in final form June 11, 1998.     

The Converse of Schur's Lemma in Noetherian Rings and Group Algebras

ABSTRACT

If M is a simple module over a ring R, then, by Schur's Lemma, its endomorphism ring is a division ring. However, the converse of this property, which we called the CSL property, does not hold in general. The object of this article is to study this converse for a few classes of rings: left Noetherian rings, V-rings and group algebras. First, we establish that a left Noetherian ring R is a CSL ring if and only if a ring R is left–artinian and primary decomposable. Secondly, we prove that a left semiartinian V-ring is CSL. At last, we study the CSL property in group algebra K [ G ] where K a field algebraically closed of characteristic p and G is a finite group of order divisible by p. Our main contribution is that K [ G ] is a CSL ring if and only if Gbf = HP where H is a normal p′-subgroup and bfP a Sylow bfp-subgroup of bfG. In this case, K [ G ] is primary decomposable.     

Rank formulae from the perspective of orthogonal projectors

For matrices F and G having the same number of rows and the orthogonal projectors P?=?FF ^? and Q?=?GG ^?, with F ^? and G ^? denoting the Moore–Penrose inverses of F and G, respectively, several formulae for ranks of various functions of F, G, P and Q are established. Besides a collection of original characterizations, many of which involve the ranks of F*G and (F?:?G) (which coincide with the ranks of PQ and P?+?Q, respectively), some properties known in the literature are reestablished in a generalized form. The variety of relationships considered shows that the approach utilized in the article, based on the partitioned representations of the projectors, provides a powerful tool of wide applicability.     

On the U-Invariant of P-Adic Function Fields

ABSTRACT

We show that a quadratic form defined over the rational function field ?(x ₁ , …, _{x n} ) of dimension at least 4.2 ⁿ  + 1 is isotropic over all fields ? _p (x ₁ , …, _{x n} ), except for finitely many primes. Partial results concerning the u-invariant of p-adic function fields are also shown.     

Multilinear polynomials with invertible values

ABSTRACT

Let 𝔅( m, n ) be a free Burnside group of a sufficiently large exponent bin with a basis of cardinality m , m  ≥  2 . We prove the existence of a free semigroup of rank 2 in Aut 𝔅( m, n ).     

Generic Global Rigidity

Suppose a finite configuration of labeled points p = (p₁,. . . ,p_n) in E^d is given along with certain pairs of those points determined by a graph G such that the coordinates of the points of p are generic, i.e., algebraically independent over the integers. If another corresponding configuration q = (q₁,. . . ,q_n) in E^d is given such that the corresponding edges of G for p and q have the same length, we provide a sufficient condition to ensure that p and q are congruent in E^d. This condition, together with recent results of Jackson and Jordán, give necessary and sufficient conditions for a graph being generically globally rigid in the plane.     

Relations between ΛBV and BV(p(n) ↑∞) Classes of Functions

The base radical class L _b(X), generated by a class X was introduced in [12]. It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, L _b(X) is the lower radical class defined by X, but otherwise X may not be contained in L _b(X). We prove that for a hereditary radical class L with semisimple class S(R), L _b(S(R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form L _b(X) are discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Spectra of degrees of some structures

We argue for the existence of structures with the spectrum {x : x ≰ a} of degrees, where a is an arbitrary low degree. Also it is stated that there exist structures with the spectrum of degrees, {x : x ≰ a} ⋃ {x : x ≰ b}, for any low degrees a and b. Supported by RFBR grant No. 05-01-00605. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 729–744, November–December, 2007.     

The Structure of the Reduced Product of Preinjective Modules Over a Hereditary Right Pure Semisimple Ring

Abstract

For a monoid S , a (left) S -act is a nonempty set B together with a mapping S ×B→B sending (s, b) to sb such that S (tb)?=?lpar;st)b and 1b ?=?b for all S , t?∈?S and B ?∈?B. Right S -acts A can also be defined, and a tensor product A ??? _s B (a set)can be defined that has the customary universal property with respect to balanced maps from A?×?B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (New York: Walter de Gruyter, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets ( S -posets). The present paper is devoted to such a generalization. A unique decomposition theorem for S -posets is given, based on strongly convex, indecomposable S -subposets, and a structure theorem for projective S -posets is given. A criterion for when two elements of the tensor product of S -posets given, which is then applied to investigate several flatness properties.     

Operators on the Lattice of Pseudovarieties of Finite Semigroups

L (F) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, Auinger, Hall and the present authors recently introduced fourteen complete congruences on L (F). Such congruences provide a framework from which to study L (F) both locally and globally. For each such congruence ρ and each U∈L (F) the ρ-class of U is an interval [U ^ρ, U _ρ]. This provides a family of operators of the form U⇒Uρ on L (F) that reveal important relationships between elements of L (F). Various aspects of these operators are considered including characterizations of U ^ρ, bases of pseudoidentities for U ^ρ, instances of commutativity (U ^ρ)^σ = U ^σ)^ρ, as well as the semigroups generated by certain pairs of such operators.     

The linear combination of vectors implies the existence of the cross and dot products

Given two vectors u and v, their cross product u × v is a vector perpendicular to u and v. The motivation for this property, however, is never addressed. Here we show that the existence of the cross and dot products and the perpendicularity property follow from the concept of linear combination, which does not involve products of vectors. For our proof we consider the plane generated by a linear combination of uand v. When looking for the coefficients in the linear combination required to reach a desired point on the plane, the solution involves the existence of a normal vector n = u × v. Our results have a bearing on the history of vector analysis, as a product similar to the cross product but without the perpendicularity requirement existed at the same time. These competing products originate in the work of two major nineteen-century mathematicians, W. Hamilton, and H. Grassmann. These historical aspects are discussed in some detail here. We also address certain aspects of the teaching of u × v to undergraduate students, which is known to carry some difficulties. This includes the algebraic and geometric denitions of u × v, the rule for the direction of u × v, and the pseudovectorial nature of u × v.     

Centrality in Rees-Sushkevich varieties

Denote by RS _n the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RS _n are called Rees-Sushkevich varieties and those that are generated by completely simple or completely 0-simple semigroups are said to be exact. For each positive integer m, define C _m RS _n to be the class of all semigroups S in RS _n with the property that if the product of m idempotents of S belongs to some subgroup of S, then the product belongs to the center of that subgroup. The classes C _m RS _n constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of C ₂ RS _n is isomorphic to the direct product of a three-element chain with the lattice of central completely simple semigroup varieties over groups of exponent dividing n. In the main result, this isomorphism is extended to include those exact varieties for which the intersection of the core with any subgroup, if nonempty, is contained in the center of that subgroup. The equational property of the varieties C _m RS _n is also addressed. For any fixed n ≥ 2, it is shown that although the varieties C _m RS _n , where m = 1, 2, ... , are all finitely based, their complete intersection (denoted by C _∞ RS _n ) is non-finitely based. Further, the variety C _∞ RS _n contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based. Received October 29, 2003; accepted in final form February 11, 2007.     

Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

We investigate the lattice of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in , which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂. Received May 25, 2000; accepted in final form November 2, 2001.     

Equations on the semidirect product of a finite semilattice by a finite commutative monoid

LetCom _t,q denote the variety of finite monoids that satisfy the equationsxy=yx andx ^t =x ^t+q . The varietyCom _1,1 is the variety of finite semilattices also denoted byJ ₁. In this paper, we consider the product varietyJ ₁*Com _t,q generated by all semidirect products of the formM*N withM∈J ₁ andN∈Com _t,q . We give a complete sequence of equations forJ ₁*Com _t,q implying complete sequences of equations forJ ₁*(Com∩A),J ₁*(Com∩G) andJ ₁*Com, whereCom denotes the variety of finite commutative monoids,A the variety of finite aperiodic monoids andG the variety of finite groups. This material is based upon work supported by the National Science Foundation under Grants No. CCR-9101800 and CCR-9300738. Many thanks to the referee for his valuable comments and suggestions.     

A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

Abstract. Let F be a family of disjoint unit balls in R ³ . We prove that there is a Helly-number n ₀ ≤ 46 , such that if every n ₀ members of F ( | F | ≥ n ₀ ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou.     

Un Nouveau ℓ (K) Qui Possede la Propriete de Grothendieck

Using the continuum hypothesis, we construct a compact spaceK such that ℓ(K) possesses the Grothendieck property, but such that the unit ball of ℓ(K)′ does not containβ N, and hence, in particular, such thatl ^∞(N) is neither a subspace nor quotient of ℓ(K). In particular,K does not contain a convergent sequence but does not containβ N.      

Homomorphisms into N *

We study the properties of continuous homomorphisms from β S into N ^* and from N ^* into N ^* . We show that the image C of N ^* under a continuous homomorphism which does not arise from the continuous extension of a homomorphism mapping N to itself, has the property that C+C is a singleton.