Subtractive Categories and Extended Subtractions

We introduce a notion of an extended operation which should serve as a new tool for the study of categories like Mal’tsev, unital, strongly unital and subtractive categories. However, in the present paper we are only concerned with subtractive categories, and accordingly, most of the time we will deal with extended subtractions, which are particular instances of extended operations. We show that these extended subtractions provide new conceptual characterizations of subtractive categories and moreover, they give an enlarged “algebraic tool” for working in a subtractive category—we demonstrate this by using them to describe the construction of associated abelian objects in regular subtractive categories with finite colimits. Also, the definition and some basic properties of abelian objects in a general subtractive category is given for the first time in the present paper. The second author acknowledges the support of Claude Leon Foundation, INTAS (06-1000017-8609) and Georgian National Science Foundation (GNSF/ST06/3-004).     

An embedding theorem for regular Mal'tsev categories

In this paper, we obtain a non-abelian analogue of Lubkin's embedding theorem for abelian categories. Our theorem faithfully embeds any small regular Mal'tsev category $\mathbb{C}$ in an n-th power of a particular locally finitely presentable regular Mal'tsev category. The embedding preserves and reflects finite limits, isomorphisms and regular epimorphisms, as in the case of Barr's embedding theorem for regular categories. Furthermore, we show that we can take n to be the (cardinal) number of subobjects of the terminal object in $\mathbb{C}$.     

Optimal Mal’tsev conditions for congruence modular varieties

For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal’tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal’tsev condition. However, the Mal’tsev condition constructed in [5] is not the simplest known one in general. Now we improve this result by constructing the best Mal’tsev condition and various related conditions. As an application, we give a particularly easy new proof of the result of Freese and Jónsson [11] stating that modular congruence varieties are Arguesian, and we strengthen this result by replacing “Arguesian” by “higher Arguesian” in the sense of Haiman [18]. We show that lattice terms for congruences of an arbitrary congruence modular variety can be computed in two steps: the first step mimics the use of congruence distributivity, while the second step corresponds to congruence permutability. Particular cases of this result were known; the present approach using TIP is even simpler than the proofs of the previous partial results.Dedicated to the memory of Ivan RivalReceived February 12, 2003; accepted in final form August 5, 2004.This revised version was published online in August 2005 with a corrected cover date.     

The Chevalley and Costant theorems for Mal’tsev algebras

Centers of universal envelopes for Mal’tsev algebras are explored. It is proved that the center of the universal envelope for a finite-dimensional semisimple Mal’tsev algebra over a field of characteristic 0 is a ring of polynomials in a finite number of variables equal to the dimension of its Cartan subalgebra, and that universal enveloping algebra is a free module over its center. Centers of universal enveloping algebras are computed for some Mal’tsev algebras of small dimensions. Supported by FAPESP grant No. 04/08537-4 and by SO RAN grant No. 1.9. Supported by FAPESP grant Nos. 05/60142-7, 05/60337-2 and by CNPq grant No. 304991/2006-6. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 560–584, September–October, 2007.     

Composites of Central Extensions Form a Relative Semi-Abelian Category

We consider trivial and central extensions, in the sense of G. Janelidze and G. M. Kelly, which are defined with respect to an adjunction between a Barr-exact category C and a Birkhoff subcategory X of C. Assuming in addition that C is a pointed Mal’tsev category with cokernels, and that X is protomodular, we prove that: (a) the class of all trivial extensions and the class of all finite composites of central extensions form relative homological category structures on C; (b) if C has finite coproducts, then the class of all finite composites of central extensions forms a relative semi-abelian category structure on C.     

Semidirect Products and Split Short Five Lemma in Normal Categories

In this paper we study a generalization of the notion of categorical semidirect product, as defined in [6], to a non-protomodular context of categories where internal actions are induced by points, like in any pointed variety. There we define semidirect products only for regular points, in the sense we explain below, provided the Split Short Five Lemma between such points holds, and we show that this is the case if the category is normal, as defined in [12]. Finally, we give an example of a category that is neither protomodular nor Mal’tsev where such generalized semidirect products exist.     

The Cuboid Lemma and Mal’tsev Categories

We prove that a regular category ? is a Mal’tsev category if and only if a strong form of the denormalised 3 × 3 Lemma holds true in ?. In this version of the 3 × 3 Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal’tsev categories (= 2-permutable) is similar to the one previously obtained for Goursat categories (= 3-permutable). We also analyse the “relative” version of the Cuboid Lemma and extend our results to that context.     

Grothendieck groups arising from contravariantly finite subcategories

The subject of the paper is the study of the relative homological properties of a given additive category C in relation to a given contravariantly finite subcategory X in C under the assumption that any X-epic has a kernel in C. We introduce the notion of the Grothendieck group relative to the pair (C, X) and also that of the Cartan map cx relative to (C, X) and we show that the cokernel of cx is isomorphic to the corresponding Grothendieck group of the stable category C/Jx We also show that if the right x-dimension of C is finite, then cx is an isomorphism. In case C is a finite dimensional k-additive Krull-Schmidt category, we introduce the notion of the x-dimension vector of an object of C. We give criteria for when an indecomposable object is determined, up to isomorphism, by its x-dimension vector.     

Browder’s Theorems through Localized SVEP

A bounded linear operator T ∈ L(X) on aBanach space X is said to satisfy “Browder’s theorem” if the Browder spectrum coincides with the Weyl spectrum. T ∈ L(X) is said to satisfy “a-Browder’s theorem” if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum. In this note we give several characterizations of operators satisfying these theorems. Most of these characterizations are obtained by using a localized version of the single-valued extension property of T. In the last part we shall give some characterizations of operators for which “Weyl’s theorem” holds.     

Relative Semi-abelian Categories

We define a relative semi-abelian category as a pair (C,E), where C is a pointed category with finite limits, and E is a class of regular epimorphisms in C satisfying certain conditions, stronger than those defining a relative homological category. Some results on the equivalence of the so-called old-style and new-style axioms for the semi-abelian categories are extended to the relative case. Partially supported by the University of Cape Town Research Associateship and Georgia National Science Foundation (GNSF/ST06/3-004).     

Categorical (binary) difference terms and protomodularity

In this paper we obtain an intrinsic syntactical characterization of protomodularity, via so-called categorical difference terms, similar to the one known in the case of varieties involving binary terms d satisfying d(x, x) = d(y, y). We also show that purely categorical modifications of the condition in the characterization give characterizations of Mal’tsev and additive categories, thus revealing a new conceptual link between these three classes of categories, and hence, also between the corresponding classes of varieties.     

On almost Chebyshev subspaces

A closed subspace F in a Banach space X is called almost Chebyshev if the set of x ε X which fail to have unique best approximation in F is contained in a first category subset. We prove, among other results, that if X is a separable Banach space which is either locally uniformly convex or has the Radon-Nikodym property, then “almost all” closed subspaces are almost Chebyshev.     

Study of some measures of dependence between order statistics and systems

Let be a random vector, and denote by X_1:n,X_2:n,…,X_n:n the corresponding order statistics. When X₁,X₂,…,X_n represent the lifetimes of n components in a system, the order statistic X_n−k+1:n represents the lifetime of a k-out-of-n system (i.e., a system which works when at least k components work). In this paper, we obtain some expressions for the Pearson’s correlation coefficient between X_i:n and X_j:n. We pay special attention to the case n=2, that is, to measure the dependence between the first and second failure in a two-component parallel system. We also obtain the Spearman’s rho and Kendall’s tau coefficients when the variables X₁,X₂,…,X_n are independent and identically distributed or when they jointly have an exchangeable distribution.     

The Change-Base Issue for Ω-Categories

Let G ： Ω→Ω＇ be a closed unital map between commutative, unital quantales. G induces a functor G^- from the category of Ω-categories to that of Ω＇-categories. This paper is concerned with some basic properties of G^-. The main results are： （1） when Ω, Ω＇ are integral, G ： Ω→Ω＇ and F ： Ω＇→Ω are closed unital maps, F is a left adjoint of G^- if and only if F is a left adjoint of G; （2） G^- is an equivalence of categories if and only if G is an isomorphism in the category of commutative unital quantales and closed unital maps; and （3） a sufficient condition is obtained for G^- to preserve completeness in the sense that GA is a complete Ω＇-category whenever A is a complete Ω-category.     

Normal Sections and Direct Product Decompositions

Abstract

In any finitely complete category, there is an internal notion of normal monomorphism. We give elementary conditions guaranteeing that a normal section s: Y → X of an arrow f: X → Y produces a direct product decomposition of the form X ? Y × W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories.     

The property of being polynomial for Mal’tsev constraint satisfaction problems

A combinatorial constraint satisfaction problem aims at expressing in unified terms a wide spectrum of problems in various branches of mathematics, computer science, and AI. The generalized satisfiability problem is NP-complete, but many of its restricted versions can be solved in a polynomial time. It is known that the computational complexity of a restricted constraint satisfaction problem depends only on a set of polymorphisms of relations which are admitted to be used in the problem. For the case where a set of such relations is invariant under some Mal’tsev operation, we show that the corresponding constraint satisfaction problem can be solved in a polynomial time. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 655–686, November–December, 2006.     

<Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> theory,L.S. category,and strong category

Relations between category and strong category are studied. The notion of a homotopy coalgebra of order r over the Ganea comonad is introduced. It is shown that cat(X) =Cat(X) holds if a finite 1-connected complex X carries such a structure with r sufficiently large.     

Cohomologically trivial internal categories in categories of groups with operations

We describe cohomologically trivial internal categories in the categoryC of groups with operations satisfying certain conditions ([15], [16]). As particular cases we obtain: ifC=Gr, H⁰(C, –)=0 iff C is a connected internal category; ifC=Ab,H ¹(C, –)=0 iff C is equivalent to the discrete internal category (Cokerd, Cokerd, 1, 1, 1, 1). We also discuss related questions concerning extensions, internal categories, their cohomology and equivalence in the categoryC.     

On countably uniform closed-spaces

Abstract

Let X be a topological space and C_c(X) be the functionally countable subalgbera of C(X). We call X to be a countably uniform closed-space, briefly, a CU C-space, if C_c(X) is closed under uniform convergence. We investigate that countably uniform closedness need not closed under finite intersection and infinite product. It is shown that if X is a countable union of quasi-components, then X is a CU C-space. We characterize C_c-embedding and also -embedding in CU C-spaces. A subset S of X is called Z_c-embedded, if each Z ∈ Z_c(S) is the restriction of a zero-set of Z_c(X). It is observed that in a zero-dimensional CU C-space, each Lindelöf subspae is Z_c-embedded. Moreover, it is shown that in CU C-spaces, each Lindelöf subspace is C_c-embedded if and only if it is c-completely separated from each zero-set, which is disjoint from it. Also in latter spaces, it is observed that for each S ? X, C_c-embedding, -embedding and Z_c-embedding coincide, when S belongs to Z_c(X) or it is a c-pseudocompact space. Finally, when X is both a CU C-space and a CP-space, then each Z_c-embedded subspace is C_c-embedded (-embedded) in X.     

Right n-angulated categories arising from covariantly finite subcategories

We define right n-angulated categories, which are analogous to right triangulated categories. Let 𝒞 be an additive category and 𝒳 a covariantly finite subcategory of 𝒞. We show that under certain conditions, the quotient 𝒞∕[𝒳] is a right n-angulated category. This has immediate applications to n-angulated quotient categories.