A note on effective descent morphisms of topological spaces and relational algebras

We formulate two open problems related to and, in a sense, suggested by the Reiterman-Tholen characterization of effective descent morphisms of topological spaces.     

Topological spaces and quasi-varieties

We describe Top ^op and Sob ^op as quasi-varieties by means of suitable schizophrenic objects.Research of the first author supported by grants from the NSERC of Canada and the FCAR du Québec. Research of the second author supported by the Topology grant 40% and by the NATO grant CRG 941330.     

Local separation in distributive semilattices

We introduce the Local Separation Property (LSP) for distributive semilattices. We show that LSP holds in many semilattices of the form Con_c A, where A is a lattice. On the other hand, we construct an abstract example of a distributive lattice without LSP. Our research is connected with the well known open problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of some lattice. Received December 10, 2004; accepted in final form June 6, 2005.     

Relative separation in distributive congruence lattices

In [5] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in Now we generalize these results using the concept of relatively separable sets (with respect to subsets) and apply them to some lattice varieties.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived November 29, 2002; accepted in final form August 19, 2004.     

Fraction-dense algebras and spaces

A fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fraction-dense f-rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For Archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fraction-dense spaces are defined as those for which C (X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover.     

A Birkhoff theorem for partial algebras via completion

An equational theory (a Birkhoff theorem) for functorial partial algebras is established via the corresponding theory for functorial total algebras.This work was done with partial support of the DFG (BRD).Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

Weakly diagonal algebras and definable principal congruences

An algebra is called weakly diagonal if every subuniverse of its square contains the graph of an automorphism. We show that every variety generated by a finite algebra with no proper subalgebras has a weakly diagonal generator. The result is applied in several ways and, in particular, to show that every arithmetical affine complete variety of finite type has equationally definable principal congruences. This paper is dedicated to Walter Taylor. Received February 22, 2005; accepted in final form June 3, 2005. Work of the first author was supported by grant No. 5368 from The Estonian Science Foundation.     

Some aspects of topological descent

The paper deals with (effective) descent morphisms for subfibrations X of the basic fibration Top/X, for topological spaces X and classes of continuous functions stable under pullback. For a category with pullbacks, we prove the stability under pullback of effective -descent morphisms for a class satisfying some suitable conditions. This plays a rôle in relating effective -descent to effective global-descent and enables us to obtain a criterion for effective étale-descent. We also show that the inclusion of the class of effective global-descent maps in the class surjective effective étale-descent is strict.Partial financial support by Centro de Matemática da Universidade de Coimbra is gratefully acknowledged.     

Topological continuous convergence

Let X be a limit space, Y a topological space. We show that c(X,Y), the limitierung of continuous convergence on LIM(X,Y), is topological whenever X is basic locally compact. For regular Y, local compactness of X is sufficient. In both cases, c(X,Y) coincides with the compact-open topology. If X satisfies a certain regularity condition, the fact that c(X,Y) is topological implies, conversely, that X is (basic) locally compact.The author would like to thank S. Weck for some inspiring discussions.     

Approximate distributive laws and finite equational bases for finite algebras in congruence-distributive varieties

For a congruence-distributive variety, Maltsev’s construction of principal congruence relations is shown to lead to approximate distributive laws in the lattice of equivalence relations on each member. As an application, in the case of a variety generated by a finite algebra, these approximate laws yield two known results: the boundedness of the complexity of unary polynomials needed in Maltsev’s construction and the finite equational basis theorem for such a variety of finite type. An algorithmic version of the construction is included. Received November 27, 1996; accepted in final form December 16, 2004.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

Topological sums and products in ZF-set theory

In ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice, the category Top of topological spaces and continuous maps is well-behaved. In particular, Top has sums (=coproducts) and products. However, it may happen that for families (Xi)_i∈I and (Yi)_i∈I with the property that each Xi is homeomorphic to the corresponding Yi neither their sums _⊕i∈IXi and _⊕i∈IYi nor their products _∏i∈IXi and _∏i∈IYi are homeomorphic. It will be shown that the axiom of choice is not only sufficient but also necessary to rectify this defect.     

Special reflexive graphs in modular varieties

We investigate a special kind of reflexive graph in any congruence modular variety. When the variety is Maltsev these special reflexive graphs are exactly the internal groupoids, when the variety is distributive they are the internal reflexive relations. We use these internal structures to give some characterizations of Maltsev, distributive and arithmetical varieties.Received August 6, 2003; accepted in final form June 4, 2004.     

Fibration of points and congruence modularity

Malcev varieties, or more generally Malcev categories, have been characterized by a property of the fibres of their fibration of points. In the same way, it is shown here that congruence modular varieties, and more generally Gumm categories, are also characterized by a property of these fibres.Received June 4, 2003; accepted in final form June 18, 2004.     

Complemented congruences on Ockham algebras

An Ockham algebra that satisfies the identity is called a K_{n, m}-algebra. Generalizing some results obtained in [2], J. Varlet and T. Blyth, in [3, Chapter 8], study congruences on K_{1, 1}-algebras. In particular, they describe the complement (when it exists) of a principal congruence and characterize these congruences that are complemented. In this paper we study the same question for K_{n, m}-algebras. Received March 24, 2005; accepted in final form April 28, 2005.     

Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.