Singular Archimedean lattice-ordered groups

W stands for the category of all archimedean l-groups with designated weak unit. The subcategory W _s of all groups with singular weak unit is analyzed as a full subcategory of W which is both epireflective and monocoreflective. A general technique for "contracting" monoreflections of a category A to a monocoreflective subcategory B is developed and then applied to W _s to show that: (i) the projectable hull in W _s is a monoreflection; (ii) essential hulls in W _s are formed by simply taking the lateral completion, and G is essentially closed in this category if and only if , where X is compact, Hausdorff and extremally disconnected; (iii) the maximum monoreflection on W _s , denoted , is obtained by contracting the maximum monoreflection on W, and G is epicomplete in W _s precisely when G is laterally -complete; (iv) the maximum essential reflection on W _s , denoted , is the contraction of the maximum essential reflection on W. Received January 22, 1997; accepted in final form June 11, 1998.     

Annihilator Primes and Foundation Primes

ABSTRACT

Let R be a Noetherian ring and M a finitely generated R -module. In this article, we introduce the set of prime ideals Fnd  M , the foundation primes of M . Using the fact that this set is nicely organized by foundation levels, we present an approach to the problem of understanding Annspec  M , the annihilator primes of M , via Fnd  M . We show: (1) Fnd  M is a finite set containing Annspec  M . Further, suppose that moreover every ideal of R has a centralizing sequence of generators; now, Annspec  M is equal to the set Ass  M of associated primes of M. Then: (2) For an arbitrary P  ∈ Fnd  M , P  ∈ Annspec  M if and only if there is no Q  ∈ Annspec  M such that P contains Q , and at the same time, the minimal foundation level on which appears P is greater than the minimal foundation level on which appears Q .     

Generating primitive positive clones

Suppose is a set of operations on a finite set A. Define PPC() to be the smallest primitive positive clone on A containing . For any finite algebra A, let PPC#(A) be the smallest number n for which PPC(CloA) = PPC(Clo _n A). S. Burris and R. Willard [2] conjectured that PPC#(A) ≤|A| when CloA is a primitive positive clone and |A| > 2. In this paper, we look at how large PPC#(A) can be when special conditions are placed on the finite algebra A. We show that PPC#(A) ≤|A| holds when the variety generated by A is congruence distributive, Abelian, or decidable. We also show that PPC#(A) ≤|A| + 2 if A generates a congruence permutable variety and every subalgebra of A is the product of a congruence neutral algebra and an Abelian algebra. Furthermore, we give an example in which PPC#(A) ≥|A| - 1)² so that these results are not vacuous. Received August 30, 1999; accepted in final form April 4, 2000.     

NIL PROBLEMS REVISITED FOR ALMOST NILPOTENT RINGS

Abstract

We begin with the notion of K-flat projectivity. For each biframe L we then introduce a binary relation ?L on it. The K-flat projective biframes are exactly such biframes with each element a of the total (first, second) part approximated by the elements x of the total (first, second) part, x?_L a and the relation ?_L being stable wrt. the meet operation on L. Further on, we introduce the notion of a K-comonad and characterize K-flat projective biframes as those biframes having a coalgebra structure for the K-comonad. The K-coherent biframes and K-flat projective biframes are coreflective in all biframes.     

A Generalized Global Cartan Decomposition: A Basic Example

Suppose G ₁ ?  G are complex linear simple Lie groups. Let  ₁ ?  be the corresponding pair of Lie algebras. For the Killing-orthogonal  of  ₁ in  we have a vector space direct sum  =  ₁⊕ , which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G ₁) = (SL (4,?),Sp (2,?)); here  =  (4,?),  ₁ =  (2,?), and  = {X ?  | X ^? = X}, where X? X ^? is the symplectic involution. We prove that G  =  G ₁exp  ∪ i G ₁exp . The key point of the proof is to study in detail the set exp ; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ^? = X, which is here a natural counterpart of the standard Pfaffian.     

Rank formulae from the perspective of orthogonal projectors

For matrices F and G having the same number of rows and the orthogonal projectors P?=?FF ^? and Q?=?GG ^?, with F ^? and G ^? denoting the Moore–Penrose inverses of F and G, respectively, several formulae for ranks of various functions of F, G, P and Q are established. Besides a collection of original characterizations, many of which involve the ranks of F*G and (F?:?G) (which coincide with the ranks of PQ and P?+?Q, respectively), some properties known in the literature are reestablished in a generalized form. The variety of relationships considered shows that the approach utilized in the article, based on the partitioned representations of the projectors, provides a powerful tool of wide applicability.     

Categorical Abstract Algebraic Logic: Local Characterization Theorems for Classes of Systems

Let ? = ?F, R, ρ? be a system language. Given a class of ?-systems K and an ?-algebraic system A = ?SEN,?N,F??, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K _A of A-systems in K as the collection of all members of K of the form 𝔄 = ? SEN,?N,F?,R ^𝔄 ?, for some set of relation systems R ^𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K, on the one hand, and local properties of K _A and global properties connecting K _A and K _A′, whenever there exists an ?-morphism ? F,α? : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K _A is an algebraic closure system, for every ?-algebraic system A, provided that K is closed under subsystems and reduced products.     

Pure ℓ-Ideals in Lattice-Ordered Rings#

An ?-ideal I of a commutative lattice-ordered ring R with positive identity element is called a pure ?-ideal if R  =  I  + ?( x ) for each x  ∈  I , where ?(x) is the ?-annihilator of x in R . In this article, we give some results on pure ?-ideals and study the ?-ideal structure of a commutative lattice-ordered ring with positive identity element by using pure ?-ideals.     

The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

We use classical results on the lattice of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice Ps (DA) of subpseudovarieties of DA, – where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. We bring forward a lattice congruence on Ps (DA), whose quotient is isomorphic to , and whose classes are intervals with effectively computable least and greatest members. Also we characterize the pro-identities satisfied by the members of an important family of subpseudovarieties of DA. Finally, letting V _k be the pseudovariety generated by the k-generated elements of DA (k≥ 1), we use all our results to compute the position of the congruence class of V _k in . Received April 24, 1996; accepted in final form April 3, 1997.     

On the Unit Groups of Commutative Modular Group Algebras of KT-Groups

ABSTRACT

In this article, we prove that the inner projection of a projective curve with higher linear syzygies has also higher linear syzygies. Specifically, if a very ample line bundle ? on a smooth projective curve X satisfies property N _p for p  ≥  1 and H ¹ (? ^? 2) =  0 , then ?( ?  q ) satisfies property N _{p ? 1} for any point q  ∈  X . We also give simple proofs of well-known theorems about syzygies and raise some questions related to the line bundles of degree 2 g which do not satisfy property N ₁ .     

Representations of algebraic groups over a 2-dimensional local field

We introduce a categorical framework for the study of representations of G(F), where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G(F), which is an object of a suitable category of representations of G(F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G(F), when we consider the action by right translations.     

Characterization of Multidimensional Stable Random Measures by Means of Vector Measures

Abstract

In connection with a symmetric α stable random measure Φ on a measurable space (F, ?) with values in R ^d , a complete metric space of symmetric finite measures on S _d?1 is constructed, and is employed to characterize the law of Φ by a unique positive measure on ? and a unique function on F × R ^d . The stochastic integral ∫ _F f d Φ is also defined for certain d × d matrix valued functions f, which for α = 2 reduces to the Wiener–Masani integral.     

Almost periodic measures on the torus

Given a skew product flow (T,T ²) on the two torus, we construct a family of flows onT ³ parametrized by elements of the circleT. We show that under a certain condition on (T,T ²) almost every flow in this family is strictly ergodic. This is used to characterize minimal subsets of the flow (T,P(T ²)) induced byT on the space of probability measures onT ². Using a result of M. Herman, we give an example to show that this characterization does not hold for everyT. To the memory of Shlomo Horowitz     

On the set of grouplikes of a coring

We focus our attention to the set Gr(■) of grouplike elements of a coring ■ over a ring A.We do some observations on the actions of the groups U(A) and Aut(■) of units of A and of automorphisms of corings of ■,respectively,on Gr(■),and on the subset Gal(■) of all Galois grouplike elements.Among them,we give conditions on ■ under which Gal(■) is a group,in such a way that there is an exact sequence of groups {1} → U(Ag) → U(A) → Gal(■) → {1},where Ag is the subalgebra of coinvariants for some g ∈ Gal(■).     

Leibniz Central Extensions on Some Infinite-Dimensional Lie Algebras

Abstract

In this paper, all one-dimensional Leibniz central extensions on the algebras of differential operators over C[t, t ^?1] and C((t)), as well as on the quantum 2-torus, the Virasoro-like algebra and its q-analog are studied. We determine all nontrivial Leibniz 2-cocycles on these infinite dimensional Lie algebras.     

On the Boundedness of Dyadic Hardy and Hardy-Littlewood Operators on the Dyadic Spaces H and BMO

Dyadic analogs of the integral Hardy and Hardy-Littlewood operators on R ₊ are introduced. It is proved that the first of them is bounded on the dyadic Hardy space H _d (R ₊), while the second one is bounded on the dyadic space BMO _d (R ₊) of functions of bounded mean oscillation on R ₊.     

Subdirect products of hereditary congruence lattices

A congruence lattice L of an algebra A is called power-hereditary if every 0-1 sublattice of Lⁿ is the congruence lattice of an algebra on Aⁿ for all positive integers n. Let A and B be finite algebras. We prove

Whaley's Theorem for Finite Lattices

Whaley's Theorem on the existence of large proper sublattices of infinite lattices is extended to ordered sets and finite lattices. As a corollary it is shown that every finite lattice L with |L|≥3 contains a proper sublattice S with |S|≥|L|^1/3. It is also shown that that every finite modular lattice L with |L|≥3 contains a proper sublattice S with |S|≥|L|^1/2, and every finite distributive lattice L with |L|≥4 contains a proper sublattice S with |S|≥3/4|L|. This revised version was published online in June 2006 with corrections to the Cover Date.     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.