Extension theorems for Fredholm and invertibility symbols

This paper is a continuation of [GK3] where the theory of Invertibility Symbol in Banach algebras was developed. In the present paper we generalize these results for the case when the Invertibility Symbol is defined on a subalgebra of the Banach algebras. The difficulty which arises here in this more general case is connected with the fact that some elements of the subalgebra may have the inverses which do not belong to the subalgebra. This generalization of the theory allows us to study the Fredholm Symbols of linear operators. Applications to subalgebras generated by two idempotents and to algebras generated by singular integral operators are presented.     

Non-standard tight closure for affine ℂ-algebras

 In this paper, non-standard tight closure is proposed as an alternative for classical tight closure on finitely generated algebras over ℂ. It has the advantage that it admits a functional definition, similar to the characteristic p definition of tight closure, where instead of the characteristic p Frobenius, its ultraproduct, the non-standard Frobenius, is used. This new closure operation cl(⋅) has the same properties as classical tight closure, to wit, (1) if A is regular, then 𝔞=cl(𝔞); (2) if A⊂B is an integral extension of domains, then cl(𝔞 B)∩A⊂cl(𝔞); (3) if A is local and is a system of parameters, then (Colon-Capturing); (4) if 𝔞 is generated by m elements, then cl𝔞 contains the integral closure of 𝔞 ^m and is contained in the integral closure of 𝔞 (Brian?on-Skoda). Received: 25 June 2002 / Revised version: 14 February 2003 Published online: 19 May 2003 Partially supported by a grant from the National Science Foundation.     

Koszulity and Koszul modules of dual extension algebras

Let A and B be algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that T is Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are Koszul. Finally, we give a necessary and sufficient condition for a dual extension algebra to have the property that all linearly presented modules are Koszul modules, which provides an effective way to construct algebras with such a property.     

Symbolic powers of monomial ideals and vertex cover algebras

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.     

Canonical extensions and ultraproducts of polarities

Jónsson and Tarski’s notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames.     

Reflection-closed varieties of multisorted algebras and minor identities

The notion of reflection is considered in the setting of multisorted algebras. The Galois connection induced by the satisfaction relation between multisorted algebras and minor identities provides a characterization of reflection-closed varieties: a variety of multisorted algebras is reflection-closed if and only if it is definable by minor identities. Minor-equational theories of multisorted algebras are described by explicit closure conditions. It is also observed that nontrivial varieties of multisorted algebras of a non-composable type are reflection-closed.     

Completely Simple Semigroups,Lie Algebras,and the Road Coloring Problem

Consider a semigroup generated by matrices associated with an edge-coloring of a strongly connected, aperiodic digraph. We call the semigroup Lie-solvable if the Lie algebra generated by its elements is solvable. We show that if the semigroup is Lie-solvable then its kernel is a right group. Next, we study the Lie algebra generated by the kernel. Lie algebras generated by two idempotents are analyzed in detail. We find that these have homomorphic images that are generalized quaternion algebras. We show that if the kernel is not a direct product, then the Lie algebra generated by the kernel is not solvable by describing the structure of these algebras. Finally, we discuss an infinite class of examples that are shown to always produce strongly connected aperiodic digraphs having kernels that are not right groups.     

Stability of representations of effective partial algebras

An algebra is effective if its operations are computable under some numbering. When are two numberings of an effective partial algebra equivalent? For example, the computable real numbers form an effective field and two effective numberings of the field of computable reals are equivalent if the limit operator is assumed to be computable in the numberings (theorems of Moschovakis and Hertling). To answer the question for effective algebras in general, we give a general method based on an algebraic analysis of approximations by elements of a finitely generated subalgebra. Commonly, the computable elements of a topological partial algebra are derived from such a finitely generated algebra and form a countable effective partial algebra. We apply the general results about partial algebras to the recursive reals, ultrametric algebras constructed by inverse limits, and to metric algebras in general. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Calabi–Yau algebras and weighted quiver polyhedra

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi–Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3-dimensional Calabi–Yau algebras can be constructed from dimer models on a torus. Toric orders are examples of a much broader class of algebras: positively graded cancellation algebras. For these algebras the CY-3 condition implies the existence of a weighted quiver polyhedron, which is an extension of dimer models obtained by replacing the torus with any two-dimensional compact orientable orbifold.     

Algebras generated by two bounded holomorphic functions

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, one of which is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension (of the closure) of such algebras. The conditions are expressed in terms of the inner part of a certain function which is explicitly derived from each pair of generators. Our results are based on identifyingz-invariant subspaces included in the closure of the algebra. The second-named author thanks the University at Albany, Harvard University, and Brown University for their hospitality during the completion of this work.     

Quasi-Hereditary Extension Algebras

The paper is a continuation of the authors' study of quasi-hereditary algebras whose Yoneda extension algebras (homological duals) are quasi-hereditary. The so-called standard Koszul quasi-hereditary algebras, presented in this paper, have the property that their extension algebras are always quasi-hereditary. In the natural setting of graded Koszul algebras, the converse also holds: if the extension algebra of a graded Koszul quasi-hereditary algebra is quasi-hereditary, then the algebra must be standard Koszul. This implies that the class of graded standard Koszul quasi-hereditary algebras is closed with respect to homological duality. Another immediate consequence is the fact that all algebras corresponding to the blocks of the category O are standard Koszul.     

Derivations and extensions of lie color algebra

In this article,the authors obtain some results concerning derivations of finitely generated Lie color algebras and discuss the relation between skew derivation space SkDer（L）and central extension H^2（L,F）on some Lie color algebras.Meanwhile,they generalize the notion of double extension to quadratic Lie color algebras,a sufficient condition for a quadratic Lie color algebra to be a double extension and further properties are given.     

Norm closure and extension of the symbolic calculus for the cone algebra

We investigate the symbolic structure of an algebra of pseudodifferential operators on manifolds with conical singularities which has been introduced by B.-W. Schulze. Our main objective is the extension of the symbolic calculus of this algebra to its norm closure in an adapted scale of Sobolev spaces. This procedure yields Banach algebras and Fréchet algebras of singular integral operators with continuous principal symbols.     

Minimally generated Boolean algebras

We introduce the class of minimally generated Boolean algebras, i.e. those algebras representable as the union of a continuous well-ordered chain of subalgebras A ₁ where A _i+1 is a minimal extension of A _i. Minimally generated algebras are closely related to interval algebras and superatomic algebras.     

A Characterization of π-Complemented Algebras

π-complemented algebras are defined as those algebras (not necessarily associative or unital) such that each annihilator ideal is complemented by other annihilator ideal. Let A be a semiprime algebra. We prove that A is π-complemented if, and only if, every idempotent in the extended centroid of A lies in the centroid of A. We also show the existence of a smallest π-complemented subalgebra of the central closure of A containing A. In the case that A is a C*-algebra, this subalgebra turns out to be a norm dense *-subalgebra of the bounded central closure of A. It follows that a C*-algebra is boundedly centrally closed if, and only if, it is π-complemented.     

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.     

Monotone clones and the varieties they determine

A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC.     

Integral Theory for Hopf Algebroids

The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.     

Natural dualities for semilattice-based algebras

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates. Presented by R. Willard.