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1.
Jorge Martínez 《Applied Categorical Structures》2011,19(2):489-504
In previous work it was shown that there is an epireflection ψ of the category of all compact normal, joinfit frames, with skeletal maps, in the full subcategory of frames which are also
strongly projectable, and that ψ restricts to the epicompletion ε, which is the absolute reflection on compact regular frames. In the first part of this paper it is shown that ψ is a monoreflection and that the reflection map is, in fact, closed. Restricted to coherent frames and maps, ψ A can then be characterized as the least strongly projectable, coherent, normal, joinfit frame in which A can be embedded as a closed, coherent, and skeletal subframe. The second part discusses the role of the nucleus d in this context. On algebraic frames with coherent skeletal maps d becomes an epireflection. Further, it is shown that e = d · ψ epireflects the category of coherent, normal, joinfit frames, with coherent skeletal maps, in the subcategory of those frames
which are also regular and strongly projectable, which are epicomplete. The action of e is not monoreflective. 相似文献
2.
Jorge Martínez 《Czechoslovak Mathematical Journal》2006,56(2):437-474
In an algebraic frame L the dimension, dim(L), is defined, as in classical ideal theory, to be the maximum of the lengths n of chains of primes p
0 < p
1 < ... < p
n
, if such a maximum exists, and ∞ otherwise. A notion of “dominance” is then defined among the compact elements of L, which affords one a primefree way to compute dimension.
Various subordinate dimensions are considered on a number of frame quotients of L, including the frames dL and zL of d-elements and z-elements, respectively. The more concrete illustrations regarding the frame convex ℓ-subgroups of a lattice-ordered group
and its various natural frame quotients occupy the second half of this exposition.
For example, it is shown that if A is a commutative semiprime f-ring with finite ℓ-dimension then A must be hyperarchimedean. The d-dimension of an ℓ-group is invariant under formation of direct products, whereas ℓ-dimension is not. r-dimension of a commutative semiprime f-ring is either 0 or infinite, but this fails if nilpotent elements are present. sp-dimension coincides with classical Krull dimension in commutative semiprime f-rings with bounded inversion. 相似文献
3.
Jorge Martínez 《Applied Categorical Structures》2012,20(2):189-208
Earlier work has shown that there is a monoreflection ψ of the category of compact normal, joinfit frames with skeletal frame maps in the subcategory consisting of strongly projectable
frames. This article extends the domain of ψ to the *-regular frames. The saturation nucleus s is a reflection with respect to weakly closed frame maps, in the subcategory of subfit frames. Moreover, s·ψ = ψ·s, on compact normal, joinfit frames with skeletal, weakly closed frame maps, and s·ψ is an epireflection, but not a monoreflection, in the subcategory of strongly projectable, regular frames, all of which are
epicomplete. 相似文献
4.
In the article [17], we introduced and investigated feebly and flatly projectable frames. In this article, we apply these
two properties to lattice-ordered groups. An example is constructed to illustrate that the two properties are distinct, which
solves a question from [17]. We also investigate these properties with respect to archimedean ℓ-groups with weak order unit,
as well as commutative semiprime f-rings. 相似文献
5.
W denotes the category of archimedean ℓ-groups with designated weak unit and ℓ-homomorphisms that preserve the weak unit. Comp denotes the category of compact Hausdorff spaces with continuous maps. The Yosida functor is used to investigate the relationship
between hull classes in W and covering classes in Comp. The central idea is that of a hull class whose hull operator preserves boundedness. We demonstrate how the Yosida functor
may be used to identify hull classes in W and covering classes in Comp. In addition, we exhibit an array of order preserving bijections between certain families of hull classes and all covering
classes, one of which was recently produced by Martínez. Lastly, we apply our results to answer a question of Knox and McGovern
about the class of all feebly projectable ℓ-groups. 相似文献
6.
Ferenc Weisz 《数学学报(英文版)》2010,26(9):1627-1640
A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W(hp, e∞) to W(Lp,e∞). This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W(L1, e∞) velong to L1. 相似文献
7.
Themba Dube 《Algebra Universalis》2018,79(1):7
For any semisimple f-ring A with bounded inversion, we show that the frame of radical ideals of A and the frame of z-ideals of A have isomorphic subfit coreflections. If we assume the Axiom of Choice, then the two coreflections are actually identical. If the f-ring has the property that the sum of two z-ideals is a z-ideal (as in the case of rings of continuous functions), then the epicompletion of the frame of z-ideals is shown to be a dense quotient of the epicompletion of the frame of radical ideals. Baer rings, exchange rings, and normal rings that lie in the class of semisimple f-rings with bounded inversion are shown to have characterizations in terms of frames of z-ideal which are similar to characterizations in terms of frames of radical ideals. 相似文献
8.
Boris Lavrič 《代数通讯》2013,41(2):1091-1096
Let Abe an Archimedean uniformly complete f-algebra with unit element. It is proved that the following conditions are equivalent:(1) Ais semi-hereditary; (2) Ais coherent; (3) Ais projectable; (4) Ais Dedekind σ-complete. 相似文献
9.
Fethi Benamor 《Journal of Mathematical Analysis and Applications》2006,322(2):599-609
Let L and M be vector lattices with M Dedekind complete, and let Lr(L,M) be the vector lattice of all regular operators from L into M. We introduce the notion of maximal order ideals of disjointness preserving operators in Lr(L,M) (briefly, maximal δ-ideals of Lr(L,M)) as a generalization of the classical concept of orthomorphisms and we investigate some aspects of this ‘new’ structure. In this regard, various standard facts on orthomorphisms are extended to maximal δ-ideals. For instance, surprisingly enough, we prove that any maximal δ-ideal of Lr(L,M) is a vector lattice copy of M, when L, in addition, has an order unit. Moreover, we pay a special attention to maximal δ-ideals on continuous function spaces. As an application, we furnish a characterization of lattice bimorphisms on such spaces in terms of weigthed composition operators. 相似文献
10.
A sup-preserving map f between complete lattices L and M is regular if there exists a sup-preserving map g from M to L such that fgf=f. In the class of completely distributive lattices, this paper demonstrates a necessary and sufficient condition for f to be regular. When L=M is a power set, our theorem reduces to the well known Zareckiĭ’s theorem which characterizes regular elements in the semigroup
of all binary relations on a set. Another application of our result is a generalization of Zareckiĭ’s theorem for quantale-valued
relations. 相似文献
11.
Mostafa Abedi 《Quaestiones Mathematicae》2018,41(6):745-760
The notion of σ?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames. 相似文献
12.
Yuri G. Zarhin 《Mathematische Zeitschrift》2006,253(3):537-554
Let ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure Ka. Let n, m ≥ 4 be integers that are not divisible by ℓ. Let f(x), h(x) ∈ K[x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set of roots of f and that Gal(h) acts doubly transitively on as well. Let J(Cf,ℓ) and J(Ch,ℓ) be the Jacobians of the superelliptic curves Cf,ℓ:yℓ=f(x) and Ch,ℓ:yℓ=h(x) respectively. We prove that J(Cf,ℓ) and J(Ch,ℓ) are not isogenous over Ka if the splitting fields of f and h are linearly disjoint over K and K contains a primitive ℓth root of unity. 相似文献
13.
Summary We consider a sequence of energy functionals for regular paths with fixed extremes and whose range is contained in a corresponding
sequence(M
h)h∈Z+ of subsets of an Hilbert space.
Assuming on eachM
h a condition similar top-convexity [C], we prove that if(M
h)h∈Z+ is convergent in the sense of Kuratowsky toM the corresponding sequence(f
h)h∈Z+of energy functionals is Γ-convergent to the functionalf relative toM and critical points off
h,i.e. the geodesics, are convergent to those off.
相似文献
14.
Nizar Marcus 《Applied Categorical Structures》2003,11(2):125-133
We present a pointfree analogue of E-compactness as introduced by Engelking and Mrówka. In particular, we show that a frame L is a closed quotient of a copower of a regular frame E iff L is complete in its E-nearness. We further show that the version of E-compactness that is more compatible with the classical one corresponds to Cauchy completeness in its E-nearness. 相似文献
15.
In this article the frame-theoretic account of what is archimedean for order-algebraists, and semisimple for people who study commutative rings, deepens with the introduction of ${\mathcal{J}}$ -frames: compact normal frames that are join-generated by their saturated elements. Yosida frames are examples of these. In the category of ${\mathcal{J}}$ -frames with suitable skeletal morphisms, the strongly projectable frames are epicomplete, and thereby it is proved that the monoreflection in strongly projectable frames is the largest such. That is news, because it settles a problem that had occupied the first-named author for over five years. In compact normal Yosida frames the compact elements are saturated. When the reverse is true one gets the perfectly saturated frames: the frames of ideals Idl(E), when E is a compact regular frame. The assignment E?Idl(E) is then a functorial equivalence from compact regular frames to perfectly saturated frames, and the inverse equivalence is the saturation quotient. Inevitable are the Yosida covers (of a ${\mathcal{J}}$ -frame L): coherent, normal Yosida frames of the form Idl(F), with F ranging over certain bounded sublattices of the saturation SL of L. These Yosida frames cover L in the sense that each maps onto L densely and codensely. Modulo an equivalence, the Yosida covers of L form a poset with a top ${\mathcal{Y}} L$ , the latter being characterized as the only Yosida cover which is perfectly saturated. Viewed correctly, these Yosida covers provide, in a categorical setting, another (point-free) look at earlier accounts of coherent normal Yosida frames. 相似文献
16.
Ferenc Weisz 《Journal of Fourier Analysis and Applications》2009,15(4):463-487
A general summability method, the so-called θ-summability method is considered for Gabor series. It is proved that if the Fourier transform of θ is in a Herz space then this summation method for the Gabor expansion of f converges to f almost everywhere when f∈L
1 or, more generally, when f∈W(L
1,ℓ
∞) (Wiener amalgam space). Some weak type inequalities for the maximal operator corresponding to the θ-means of the Gabor expansion are obtained. Hardy-Littlewood type maximal functions are introduced and some inequalities are
proved for these. 相似文献
17.
There are many results proved using the Axiom of Choice. Using point-free topology, we can prove some of these results without
using this axiom. B. Banaschewski in [Pointfree Topology and the Spectra of f-rings, Ordered algebraic structures (Curacoa, 1995), Kluwer, Dordrecht, 123–148], studying the spectra of f-rings, describes the point-free version of the classical Gelfand duality without using the Axiom of Choice In this paper,
referring to [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct.
12 (2004), 397–409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of ℓ-Modules, To appear in J. Pure Appl. Algebra], we describe a point-free version of the classical Kakutani duality. For this, using one of the spectra given in [Ebrahimi,
M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct.
12 (2004), 397–409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of l-Modules, To appear in J. Pure Appl. Algebra], we find an adjunction between the category of compact completely regular frames with
frame maps and the category of Archimedean bounded Riesz spaces with continuous Riesz maps. 相似文献
18.
Bernhard Banaschewski 《Journal of Pure and Applied Algebra》2010,214(12):2159-2164
This paper shows that the compact completely regular coreflection in the category of frames is given by the frame of Jacobson radical ideals of the ring RL of real-valued continuous functions on L, as an alternative to its familiar representations in terms of (i) the l-ideals of RL as lattice-ordered ring or (ii) the ideals of the bounded part of RL which are closed in the usual uniform topology. Further, in analogy with this, the compact zero-dimensional coreflection will also be described in terms of ring ideals, this time of the ring ZL of integer-valued continuous functions on L. 相似文献
19.
John Frith 《Quaestiones Mathematicae》2016,39(1):115-134
A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames.We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames.A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames. 相似文献
20.
Wolfgang Rump 《Applied Categorical Structures》2009,17(2):153-174
We give a axiomatic treatment of the absolute in the category of all topological spaces and characterize it as a cover with
respect to the full subcategory of extremally disconnected spaces. As an application, we obtain the strongly projectable hull
of an abelian l-group as a unique lifting of the absolute of its spectrum.
Dedicated to B. V. M. 相似文献