Inj0在Top上的Monadicity

本文主要证明了全体内射Ｔ₀－空间及强代数映射构成的范畴Ｉｎｊ₀恰是Ｅｉｌｅｎｂｅｒｇ－Ｍｏｏｒｅ范畴Ｔｏｐ^Ｔ,这里Ｔ是Ｔｏｐ与Ｓｌａｔ之间的一对偶伴随导出的ｍｏｎａｄ,由此推出Ｉｎｊ₀在Ｔｏｐ上是ｍｏｎａｄｉｃ的．     

Injective Hulls in POSV

We determine the injective objects and hulls in the category POS_V. This category is similar to the one of join semilattices but contains all partially ordered sets. The results of this paper have applications, for instance, in the theory of (generalized) ultrametric spaces.     

Injectivity of topological categories

Injective objects in concrete categories frequently turn out to be objects with particularly pleasant properties. Often some form of completeness provides a characterization of injectivity in such a category, with injective hulls achieved through certain standard completion processes. Several results during the past decade have shown that certain specific topological categories are precisely the injective objects in various natural quasicategories of concrete categories, with injective hulls obtained via certain sieve constructions. When the base category is trivial, some of these results specialize to classical results in certain categories of ordered structures; e.g., the injectives in posets characterized as complete lattices, with injective hulls the MacNeille completions, and the injectives in semilattices characterized as locales, with injective hulls the locale hulls.This paper contains two main results. The first provides a characterization of injective objects in a setting sufficiently general as to include the above mentioned characterizations as well as many others. The second theorem gives a characterization of those objects that have injective hulls, and provides a construction of the hulls as well. Corollaries of this theorem yield numerous known injective hull constructions. The second theorem uses a much stronger hypothesis than the first. That this hypothesis is indispensible follows from a result of E. Nelson on the non-existive of injective hulls of certain -semilattices.In Memory of Evelyn NelsonPresented by F.E.J. Linton.This research was partially sponsored by the U.S. National Science Foundation Grant DCR-8604080 and by support from the National Academies of Sciences of Czechoslovakia and the United States.     

CARTESIAN CLOSED TOPOLOGICAL HULLS AS INJECTIVE HULLS

Abstract

It is shown that (concretely) Cartesian closed topological hulls can be characterized as injective hulls in a rather natural setting. The characterization of locale hulls as injective hulls in the category of (meet-) semilattices by Bruns & Lakser and Born & Kimura constitutes a special case.     

On unitary down-closed regular monomorphisms of pomonoid actions

Abstract

In this paper we characterize injective objects in the category of S-posets and S-poset maps for a pomonoid S, with respect to the class of unitary down-closed embeddings. Also, the behaviour of this notion of injectivity with respect to products and coproducts is studied. Then we introduce the notion of weakly regular d-injectivity in arbitrary slices of the category of S-posets, which is applied to investigate the Baer criterion. Finally we present an example to show that these objects are not regular injective, in general.     

Rogers Semilattices of Finite Partially Ordered Sets

It is proved that the principal sublattice of a Rogers semilattice of a finite partially ordered set is definable. For this goal to be met, we present a generalization of the Denisov theorem concerning extensions of embeddings of Lachlan semilattices to ideals of Rogers semilattices.     

域上三角矩阵空间保幂等与立方幂等的加法单映射

本文刻划了特征不为2的域上三角矩阵空间保幂等加法单映射，并由此获得了特征不为2及3的域上三角矩阵空间保立方幂等加法单映射的形式．     

Non-Complementedness and Non-Distributivity of Kleene Degrees

In this note, we study the complementedness and the distributivity of upper semilattices of Kleene degrees assuming V = L. K denotes the upper semilattice of all Kleene degrees. We prove that if V = L, then some sub upper semilattices of K are non-complemented and some are non-distributive.     

Injective Rings

The purpose of this article is to determine the injective objects in some complete categories of rings. All rings are assumed to have identities and it is assumed that the homomorphisms preserve these identities. We recall that an object Q in a category is called injective if for every diagram where A′ → A is a monomorphism, there is a map A → Q making the triangle commute. The zero ring belongs to all the categories discussed and it is easy to see that it is an injective object. For the categories of commutative rings, strongly regular and commutative regular rings we show that the zero ring is the only injective by using the fact that an injective object must be a retract of any extension. We include in this section the known results which characterize the injective rings and p-rings. The second part of the paper discusses injectivity with respect to regular monomorphisms. Some necessary categorical background is given and it is then shown that results analagous with those of the first section hold (including the known Boolean and p-ring cases). In an abelian category all monomorphisms are regular, so in the study of the injective objects, for example injective modules, there are not two separate cases.     

On the definition of a Lachlan semilattice

We study the algorithmic properties of the semilattices introduced in 1972 by Lachlan in his work on recursively enumerable m-degrees, the so-called Lachlan semilattices. We show that in Lachlan’s definition the effectivity condition on the meet can be omitted in the sequence determining such a semilattice.     

Maximal (semi-)lattices of fractions and injective hulls

We show that the maximal (semi-)lattice of fractions of a (semi-)lattice may be obtained as a canonical subsemilattice of the bicommutator of an injective hull of the (semi-)lattice under consideration, whithin a suitably defined category of semilattices. The construction is roughly similar to that of a maximal semigroup of quotients, but there are significant differences due to the commutativity and idempotency of (semi-)lattice operations. In particular, the construction is effective in the sense that it avoids the use of Zorn's lemma to obtain the required injective hulls.     

Subdirectly irreducible sectionally pseudocomplemented semilattices

Sectionally pseudocomplemented semilattices are an extension of relatively pseudocomplemented semilattices—they are meet-semilattices with a greatest element such that every section, i.e., every principal filter, is a pseudocomplemented semilattice. In the paper, we give a simple equational characterization of sectionally pseudocomplemented semilattices and then investigate mainly their congruence kernels which leads to a characterization of subdirectly irreducible sectionally pseudocomplemented semilattices. Supported by the Council of the Czech Government, MSM 6198959214.     

Injective and projective objects in the category of locally compact modules over the ring of integral values of a global field

In the paper injective and projective objects in the category of locally compact modules over the ring of integral values of a global field are described together with the objects of this category possessing injective and projective resolvents. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 118–123, July, 1997. Translated by A. I. Shtern     

The Dedekind MacNeille site completion of a meet semilattice

The cuts of the classical Dedekind-MacNeille completion DM(S) of a meet semilattice S give rise to a natural cut coverage in the down-set frame \({\mathcal{D}S}\): down-set D covers element s if s lies below all upper bounds of D. This, in turn, leads to what we call the Dedekind-MacNeille frame extension DMF(S). The meet semilattices S for which DM(S) = DMF(S), which we refer to as proHeyting semilattices, can be specified by a simple formula, and we provide a number of equivalent characterizations. A sample result is that DM(S) = DMF(S) iff DM(S) is a Heyting algebra iff DM(S) coincides with the Bruns-Lakser injective envelope.     

Fibrewise injectivity and Kock–Zöberlein monads

Using Escardó’s characterization of injectivity via Kock–Zöberlein monads, we introduce suitable monads in comma categories of topological spaces that yield characterizations of fibrewise injectivity in topological T0-spaces, with respect to the class of embeddings, and of dense, of flat and of completely flat embeddings. Characterizations, in the category of topological spaces, of injective maps with respect to the same classes of embeddings follow easily from the results obtained for T0-spaces. Moreover, it is shown that, together with the corresponding embeddings, injective continuous maps form a weak factorization system in the category of topological (T0-)spaces and continuous maps.     

一类逆半群的亚直可约性

本文利用逆半群上的同余扩张，讨论了一类逆半群的亚直可约性，并刻划了这类逆半群的幂等元集的特征。     

The universal Lachlan semilattice without the greatest element

We deal with some upper semilattices of m-degrees and of numberings of finite families. It is proved that the semilattice of all c.e. m-degrees, from which the greatest element is removed, is isomorphic to the semilattice of simple m-degrees, the semilattice of hypersimple m-degrees, and the semilattice of Σ ₂ ⁰ -computable numberings of a finite family of Σ ₂ ⁰ -sets, which contains more than one element and does not contain elements that are comparable w.r.t. inclusion. Supported by the Grant Council (under RF President) for Young Russian Scientists via project MK-1820.2005.1. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 299–345, May–June, 2007.     

Current algebras, highest weight categories and quivers

We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametrized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on the affine weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to an interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and star-shaped type, as well as tame quasi-hereditary algebras, that arise from our study.