On compact quantum semigroup QS red

We study some properties of a reduced semigroup C*-algebra of a semigroup S. For the semigroup C*-algebra generated by the deformation of the algebra of continuous functions on a compact abelian group we obtain a structure of a compact quantum semigroup. We also consider morphisms of constructed compact quantum semigroups.     

Classes of abelian groups related to sequential convergence

We study classes of abelian groups related to sequential com¬pactness and its generalizations (completeness, coarseness and sequential pre-compactness) in convergence groups. In particular, we describe the algebraic structure of the abelian groups on which every coarse convergence is complete and we prove that: i) every abelian group admits a sequentially precompact convergence; ii) every algebraically compact abelian group admits a sequen¬tially compact convergence.     

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a Euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.     

Algebraic endomorphisms of LCA-groups that preserve closed subgroups

The object of this paper is the classification of those algebraic (i.e. not necessarily continuous) endomorphisms of a locally compact abelian group leaving invariant all closed subgroups. In a canonical way they turn out to form again a locally compact abelian group which can be determined up to isomorphism. If the group is totally disconnected or not periodic all endomorphisms with this property are continuous and form a topological ring.     

Invariant Measures in Free MV-Algebras

MV-algebras can be viewed either as the Lindenbaum algebras of ?ukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free _n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0, 1] ⁿ . The maximal spectrum of Free _n is canonically homeomorphic to [0, 1] ⁿ , and the automorphisms of the algebra are in 1–1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this article, we prove that the only probability measure on [0, 1] ⁿ which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure.     

Abstract Logics, Logic Maps, and Logic Homomorphisms

What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics? In order to give satisfactory answers we generalize and further develop the topological approach of [4] and present the foundations of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines a topology on the set of theories. We develop a theory of logic maps and show in what way they induce (continuous, open) functions on the corresponding topological spaces. We also establish connections to well-known notions such as translations of logics and the satisfaction axiom of institutions [5]. Logic homomorphisms are maps that behave in some sense like continuous functions and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [2]. This research was supported by the grant CNPq/FAPESB 350092/2006-0.     

The Rohlin tower theorem and hyper-finiteness for actions of continuous groups

We prove the Rohlin tower theorem for free measure preserving actions of locally compact second countable solvable groups and almost connected amenable groups. This theorem was known for l.c.s.c. abelian groups and was recently extended by Ornstein and Weiss to discrete solvable groups. We extend their methods to the continuous case, using the structure theory of the class of groups under consideration. As a corollary we obtain that free actions of such groups generate hyperfinite equivalence relations. Work supported in part by NSF grant MCS 74-19876. A02.     

Effectiveness in RPL, with applications to continuous logic

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of ?ukasiewicz logic) and continuous logic, and prove effective versions of some related theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; we show that provability degree of a formula with respect to a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory (such as that of a probability space or an L^p Banach lattice) is decidable.     

On Moduli Spaces for Abelian Categories

We show that if 𝒜 is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.     

Preservation theorems in linear continuous logic

Linear continuous logic is the fragment of continuous logic obtained by restricting connectives to addition and scalar multiplications. Most results in the full continuous logic have a counterpart in this fragment. In particular a linear form of the compactness theorem holds. We prove this variant and use it to deduce some basic preservation theorems.     

Almost-invertible Toeplitz operators and K-theory

We consider the question of when a Toeplitz operator with continuous symbol on a connected compact abelian group is almost invertible, and show that this occurs precisely when the symbol is invertible and has zero topological index. The proof uses someK-theory computations.     

General Heart Construction on a Triangulated Category (I): Unifying <Emphasis Type="Italic">t</Emphasis>-Structures and Cluster Tilting Subcategories

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any t-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories, when the cotorsion pair comes from a cluster tilting subcategory, or a t-structure, respectively.     

Continuous L-domains in logical form

We introduce a framework of approximable disjunctive propositional logic, which is the logic that results from a disjunctive propositional logic by adding an additional connective. The Lindenbaum algebra of this logic is an approximable dD-algebra. We show that for any approximable dD-algebra, its approximable filters ordered by set inclusion form a continuous L-domain. Conversely, every continuous L-domain can be represented as an approximable dD-algebra. Moreover, we establish a categorical equivalence between the category of approximable dD-algebras with approximable dD-algebra morphisms and that of continuous L-domains with Scott-continuous functions. This extends Abramsky's Domain Theory in Logical Form to the world of continuous L-domains. As an application, we give an affirmative answer to an open problem of Chen and Jung.     

Bernoulli automorphisms of finitely generated free MV-algebras

MV-algebras can be viewed either as the Lindenbaum algebras of ?ukasiewicz infinite-valued logic, or as unit intervals [0,u] of lattice-ordered abelian groups in which a strong order unit u>0 has been fixed. They form an equational class, and the free n-generated free MV-algebra is representable as an algebra of piecewise-linear continuous functions with integer coefficients over the unit n-dimensional cube. In this paper we show that the automorphism group of such a free algebra contains elements having strongly chaotic behaviour, in the sense that their duals are measure-theoretically isomorphic to a Bernoulli shift. This fact is noteworthy from the viewpoint of algebraic logic, since it gives a distinguished status to Lebesgue measure as an averaging measure on the space of valuations. As an ergodic theory fact, it provides explicit examples of volume-preserving homeomorphisms of the unit cube which are piecewise-linear with integer coefficients, preserve the denominators of rational points, and enjoy the Bernoulli property.     

Order continuity from a topological perspective

Positivity - We study three types of order convergence and related concepts of order continuous maps in partially ordered sets, partially ordered abelian groups, and partially ordered vector...     

Hochschild cohomology of abelian categories and ringed spaces

This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer-Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf.     

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.     

The Banach algebras generated by representations of abelian semigroups

Let S be a suitable subsemigroup of a locally compact abelian group. We consider the class of Banach algebras A for which there exists a continuous homomorphism ω : L ¹ (S) → A with dense range. We study the behavior of the radical in such algebras. As an application, we present some results concerning the structure of weakly compact homomorphisms of semigroup algebras.