ANOTHER CHARACTERIZATION OF BICOREFLECTIVE SUBCATEGORIES

Abstract

In this paper, the relation between the notion of a discrete functor (see [4]) and the notion of a fine functor (see [1]) is examined. As a generalization of the notion of a F-fine object (see [1]), discrete functors T: A → X are used to define K-fine objects, where K is a class of A-objects. It is shown that if T is in addition semi-topological, then (as for F-fine objects in a topological category, see [1]) the class of K-fine objects determines a bicoreflective subcategory of A. Moreover, it is shown that in co-complete, co-(well-powered) categories, the existence of bicoreflective subcategories is equivalent to the existence of functors that are both discrete and semi-topological.     

Adjoining an Identity to a Reduced Archimedean f-Ring

Let frA denote the category of f-rings which are reduced and Archimedean, and let Φ be the (nonfull) subcategory of such rings with identity (each with the natural morphisms). Some time ago, the second author showed, using his representation theory, that for each A ∈ | frA| there is a certain minimal embedding u _A :A→ uA ∈ | Φ|. More recently, he has revisited the representation theory, expanding it to include the representation of morphisms. Based upon this, the present article analyzes the operator u:| frA| → Φ: the construction of uA is tidied, several characterizations of the pair (u _A , uA) are given, and the relation between the maximal ideal structures of A and uA is described. Membership in the class U of frA-morphisms that are “u-extendable” is characterized and it is shown that U = (| frA|,U) is a category in which Φ is a full essentially-reflective subcategory. The frA-objects are characterized for which, respectively, ? B(frA(A, B) = U (A, B)), and, ? B ≠ 0(frA(B, A) = U(B, A)).     

Freely adjoining a relative complement to a lattice

Let K be a lattice, and let a < b < c be elements of K. We adjoin freely a relative complement u of b in [a, c] to K to form the lattice L. For two polynomials A and B over K ∪ {u}, we find a very simple set of conditions under which A and B represent the same element in L, so that in L all pairs of relative complements in [a, c] can be described. Our major result easily follows: Let [a, c] be an interval of a lattice K; let us assume that every element in [a, c] has at most one relative complement. Then K has an extension L such that [a, c] in L, as a lattice, is uniquely complemented.As an immediate consequence, we get the classical result of R. P. Dilworth: Every lattice can be embedded into a uniquely complemented lattice. We also get the stronger form due to C. C. Chen and G. Grätzer: Every at most uniquely complemented bounded lattice has a {0, 1}-embedding into a uniquely complemented lattice. Some stronger forms of these results are also presented.A polynomial A over K ∪ {u} naturally represents an element 〈A 〉 of L. Let us call a polynomial A minimal, if it is of minimal length representing x. We characterize minimal polynomials.Dedicated to the memory of Ivan RivalReceived February 12, 2003; accepted in final form June 18, 2004.This revised version was published online in August 2005 with a corrected cover date.     

DISCRETE FUNCTORS

Abstract

The concept of a T-discrete object is a generalization of the notion of discrete spaces in concrete categories. In this paper. T-discrete objects are used to define discrete functors. Characterizations of discrete functors are given and their relation to other important functors are studied. A faithful functor T: A → X is discrete iff the full subcategory B of A consisting of all T-discrete objects is (X-iso)-coreflective in A. It follows that the existence of bicoreflective subcategories is equivalent to the existence of suitable discrete functors. Finally, necessary and sufficient conditions are found such that for a given functor T: A → X, the full subcategory B of A consisting of all T-discrete A-objects is monocoreflective in A.     

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.     

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

Some Results on Injective Modules Over a Ring of Krull Type

ABSTRACT

In this article, we study injective modules over a ring of Krull type A. Our main result is E(K/A)? ?_{ω∈Ω ^t} E(K/?_ω), where Ω ^t is a thin defining family of valuations of A. We also characterize the rings of Krull type A such that TE(K/A) is a cogenerator of the quotient category Mod(A)/?₀, where ?₀ is the thick subcategory of the modules with trivial maps into the codivisorial modules.     

Algebras with a compatible uniformity

Given a variety of algebras V, we study categories of algebras in V with a compatible structure of uniform space. The lattice of compatible uniformities of an algebra, Unif A, can be considered a generalization of the lattice of congruences Con A. Mal'cev properties of V influence the structure of Unif A, much as they do that of Con A. The category V[CHUnif] of complete, Hausdor. such algebras in the variety V is particularly interesting; it has a factorization system , and V embeds into V[CHUnif] in such a way that is the subcategory of onto and the subcategory of one-one homomorphisms. Received February 17, 2000; accepted in final form April 1, 2001.     

A uniqueness theorem from partial transmission eigenvalues and potential on a subdomain

It is well known that a spherically symmetric wave speed problem in a bounded spherical region may be reduced, by means of Liouville transform, to the Sturm–Liouville problem L(q) in a finite interval. In this work, a uniqueness theorem for the potential q of the derived Sturm–Liouville problem L(q) is proved when the data are partial knowledge of the given spectra and the potential. Copyright © 2016 John Wiley & Sons, Ltd.     

On Hereditary Coreflective Subcategories of Top

Let A be a topological space which is not finitely generated and CH(A) denote the coreflective hull of A in Top. We construct a generator of the coreflective subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a prime space and has the same cardinality as A. We also show that if A and B are coreflective subcategories of Top such that the hereditary coreflective kernel of each of them is the subcategory FG of all finitely generated spaces, then the hereditary coreflective kernel of their join CH(AB) is again FG.     

Q-universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type 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.? It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A _i : i < 2ω) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal. Received July 19, 2001; accepted in final form July 11, 2002.     

Finitely arithmetically fixed elements of a field

An element r of a field L will be called finitely arithmetically fixed (f.a.f.) if there exists some finite subset A of L containing r such that every map f from A to L which behaves “like a homomorphism” on A, leaves r fixed. This notion will be generalized to a relative one for any field extension L/K, and several results describing the set of f.a.f. elements are obtained. Received: 21 December 2005     

The hereditariness of the upper radical

Summary Starting from a regular classM, one can construct the upper radicalU M of the classM in a category which is like that of associative, alternative or not necessarily associative rings, or that of Lie rings. It turns out that in quite a few cases the upper radical is hereditary. (cf.Suliski [7], Rjabuhin [6], Armendariz [2], Szász—Wiegandt [8]).W. G. Leavitt has suggested the problem: Give a necessary and sufficient condition to be satisfied by the regular classM so that the upper radical classU M ofM is hereditary. In the present paper we shall give such a necessary and sufficient condition. If the classM satisfies an even stronger condition, then theU M-semisimple objects are subdirectly embeddable in a (direct) product ofM-objects. Also a necessary and sufficient condition is given which assures that eachU M-semisimple object can be subdirectly embedded in a (direct) product ofM-objects.This work was done when the second named author was in the University of Islamabad under UNESCO-UNDP Special Fund Pak. 47.     

The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸?, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸?.     

Inverse Sturm–Liouville problem on equilateral regular tree

In this article, we consider a spectral problem generated by the Sturm–Liouville equation on the edges of an equilateral regular tree. It is assumed that the Dirichlet boundary conditions are imposed at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. The potential in the Sturm–Liouville equations, the same on each edge, is real, symmetric with respect to the middle of an edge and belongs to L ₂(0,?a) where a is the length of an edge. Conditions are obtained on a sequence of real numbers necessary and sufficient to be the spectrum of the considered spectral problem.     

NIL PROBLEMS REVISITED FOR ALMOST NILPOTENT RINGS

Abstract

We begin with the notion of K-flat projectivity. For each biframe L we then introduce a binary relation ?L on it. The K-flat projective biframes are exactly such biframes with each element a of the total (first, second) part approximated by the elements x of the total (first, second) part, x?_L a and the relation ?_L being stable wrt. the meet operation on L. Further on, we introduce the notion of a K-comonad and characterize K-flat projective biframes as those biframes having a coalgebra structure for the K-comonad. The K-coherent biframes and K-flat projective biframes are coreflective in all biframes.     

On linear versions of some addition theorems

Let K ? L be a field extension. Given K-subspaces A, B of L, we study the subspace ?AB? spanned by the product set AB = {ab∣ a ∈ A, b ∈ B}. We obtain some lower bounds on dim _K ?AB? and dim _K ?B ⁿ ? in terms of dim _K A, dim _K B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.     

Gelfand-Kirillov dimension under base field extension

LetF ⊂K be a field extension,A be aK-algebra. It is proved that, in general, GK dim _F A≥GK dim _K A+tr _F (K). For commutative algebras or Noetherian P.I. algebras, the equality holds. Two examples are also constructed to show that: (i) there exists an algebraA such that GK dim _F A=GK dim _K A+tr _F (K)+1; (ii) there exists an algebraic extensionF ⊂K and aK-algebraA such that GK dim _F A=∞, but GK dim _K A<∞.     

Minimal shells containing a convex surface in Minkowski space

Summary LetK be the unit ball of a Minkowski space (finite dimensional Banach space). AK-shell is the closed set of all points between two concentric balls of the space. We consider different assignments of size to aK-shell and investigate theK-shells with minimum size which contain a given convex surface. Our results extend to Minkowski geometry classical results on minimal shells in Euclidean space. This article was processed by the author using the LATEX style file from Springer-Verlag.