Non-Pointed Strongly Protomodular Theories

We give criterions for strong protomodularity and prove that the strong protomodularity of an algebraic theory is inherited by its models in a category with finite limits. We give examples of strongly protomodular theories with several constants: C ^*-algebras, rings, Heyting algebras and Boolean algebras.     

On subtractive varieties,I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a classical ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.Presented by A. F. Pixley.     

The overlap algebra of regular opens

Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, overlap algebras offer a negation-free framework suitable, among other things, for the development of point-free topology. A lot of topology can be done “inside” the language of overlap algebra. In particular, we prove that the collection of all regular open subsets of a topological space is an example of overlap algebra which, under natural hypotheses, is atomless. Since they are a constructive counterpart to complete Boolean algebras and, at the same time, they have a more powerful axiomatization than Heyting algebras, overlap algebras are expected to turn out useful both in constructive mathematics and for applications in computer science.     

Some properties of lusztig's isomorphism

It is a longstanding open problem in algebraic model theory to determine the model companions of the varieties of relative Stone algebras. Following Weispfenning's general model theory of Boolean products of structures we obtain various theories of Heyting algebras which are model and substructure complete. This works by adding only finitely many constant symbols to the language of Heyting algebras, one of which denoting a global dual atom. Thereby we especially obtain quantifier elimination for theories of atomless Post algebras of order n.     

Cohomological reduction by split pairs

A new method is developed to compare cohomology in module categories of different rings. This method does in general not produce isomorphisms, but surjective (or injective) maps between extension groups of modules over the two rings involved. Applications of this method are given to abstract problems—we recover and extend results on the strong no loops conjecture—and to algebras naturally coming up in invariant theory—we relate the cohomology of Brauer algebras with that of various symmetric groups.     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

Exponentiation for unitary structures

Motivated by the observation that both pretopologies and preapproach limits can be characterized as those convergence relations which have a unit for a suitable composition, we introduce the category Algu(T;V) of reflexive and unitary lax algebras, for a symmetric monoidal closed lattice V and a Set-monad T=(T,e,m). For T=U the ultrafilter monad, we characterize exponentiable morphisms in Algu(U;V). Further, we give a sufficient condition for an object to be exponentiable in the category Alg(U;V) of reflexive and transitive lax algebras. This specializes to known and new results for pretopological, preapproach and approach spaces.     

Measures on minimally generated Boolean algebras

We investigate properties of minimally generated Boolean algebras. It is shown that all measures defined on such algebras are separable but not necessarily weakly uniformly regular. On the other hand, there exist Boolean algebras small in terms of measures which are not minimally generated. We prove that under CH a measure on a retractive Boolean algebra can be nonseparable. Some relevant examples are indicated. Also, we give two examples of spaces satisfying some kind of Efimov property.     

Topo-canonical completions of closure algebras and Heyting algebras

We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski compactifications, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show that ideal completions preserve no identities of Heyting algebras. We also characterize definable classes of topological spaces. Received January 20, 2006; accepted in final form September 12, 2006.     

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.     

Are there convenient subcategories of Top?

The category Top has several pleasant properties but fails to have other desirable ones. Consequently there have been various attempts to replace Top by more convenient categories; mostly subcategories or supercategories of Top. Whereas several of the supercategories of Top have extremely pleasant properties, all the subcategories of Top investigated so far have some deficiencies. In the present article it is shown why this is so and why the search for more convenient subcategories of Top has to be in vain. As a preparation in Section 1 nine convenience-properties for topological categories are defined and their mutual relations are analyzed. In Section 2 it is shown that every topological subcategory of Top (under some minor, natural assumptions) it-with respect to each of the 9 mentioned properties-as convenient or inconvenient as Top itself.     

Ordered topological structures

The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general level of lax algebras, so that our categories do not concern just ordered topological spaces, but also sets with two interacting orders, approach spaces with an additional metric, etc.     

On Flat Semimodules over Semirings

The following analog of the characterization of flat modules has been obtained for the variety of semimodules over a semiring R: A semimodule _RA is flat (i.e., the tensor product functor – A preserves all finite limits) iff A is L-flat (i.e., A is a filtered colimit of finitely generated free semimodules). We also give new (homological) characterizations of Boolean algebras and complete Boolean algebras within the classes of distributive lattices and Boolean algebras, respectively, which solve two problems left open in [14]. It is also shown that, in contrast with the case of modules over rings, in general for semimodules over semirings the notions of flatness and mono-.atness (i.e., the tensor product functor – A preserves monomorphisms) are different.     

Injective spaces via adjunction

The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.     

并素元有限生成格的弱直积分解

本文建立了并素元有限生成格的弱直积分解,并给出一个解决并素元生成的完全Heyting代数的直积分解问题的新方法;作为弱直积分解的应用,证明了并素元有限生成的完全Heyting代数必然同构于有限个既约的完全Heyting代数的直积,证明了并素元有限生成格是Boole代数的充要条件是它同构于某有限集的幂集格.     

Cube-like structures generated by filters

The join semi-lattice of faces of an n-cube has a rich structure. In considering generalizations of these structures we are led to looking at interval algebras constructed using Boolean filters. We look at the structure of these algebras and their automorphism groups.Dedicated to the memory of Gian-Carlo Rota     

Topological theories and closed objects

Recent work of several authors shows that many categories of interest to topologists can be represented as categories of lax algebras. In this paper we introduce the concept of a topological theory as a syntactical tool to deal with lax algebras, and show the usefulness of our approach by applying it to the study of function spaces.     

The enriched Vietoris monad on representable spaces

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X   is totally cocomplete if and only if X^op$X^{\mathrm{op}}$ is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.