Commutators and decompositions of orthomodular lattices

We show that in any complete OML (orthomodular lattice) there exists a commutatorc such that [0,c ] is a Boolean algebra. This fact allows us to prove that a complete OML satisfying the relative centre property is isomorphic to a direct product [0,a] × [0,a ] wherea is a join of two commutators, [0,a] is an OML without Boolean quotient and [0,a ] is a Boolean algebra. The proof uses a new characterization of the relative centre property in complete OMLs. In a final section, we specify the previous direct decomposition in the more particular case of locally modular OMLs.     

Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

Quasicoherent sheaves on complex noncommutative two-tori

We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus T as an ind-object in the category of holomorphic vector bundles on T. Extending the results of [10] and [9] we prove that the derived category of quasicoherent sheaves on T is equivalent to the derived category of usual quasicoherent sheaves on the corresponding elliptic curve. We define the rank of a quasicoherent sheaf on T that can take arbitrary nonnegative real values. We study the category Qcoh(η _T ) obtained by taking the quotient of the category of quasicoherent sheaves by the subcategory of objects of rank zero (called torsion sheaves). We show that projective objects of finite rank in Qcoh(η _T ) are classified up to an isomorphism by their rank. We also prove that the subcategory of objects of finite rank in Qcoh(η _T ) is equivalent to the category of finitely presented modules over a semihereditary algebra.     

Orthocomplemented lattices with a symmetric difference

Modelling an abstract version of the set-theoretic operation of symmetric difference, we first introduce the class of orthocomplemented difference lattices (). We then exhibit examples of ODLs and investigate their basic properties finding, for instance, that any ODL induces an orthomodular lattice (OML) but not all OMLs can be converted to ODLs. We then analyse an appropriate version of ideals and valuations in ODLs and show that the set-representable ODLs form a variety. We finally investigate the question of constructing ODLs from Boolean algebras and obtain, as a by-product, examples of ODLs that are not set-representable but that “live” on set-representable OMLs. Received April 10, 2007; accepted in final form February 12, 2008.     

On split Lie triple systems II

In [4] it is studied that the structure of split Lie triple systems with a coherent 0-root space, that is, satisfying [T ₀, T ₀, T] = 0 and [T ₀, T _α , T ₀] ≠ 0 for any nonzero root α and where T ₀ denotes the 0-root space and T _α the α-root space, by showing that any of such triple systems T with a symmetric root system is of the form T = U + Σ _j I _j with U a subspace of the 0-root space T ₀ and any I _j a well described ideal of T, satisfying [I _j , T, I _k ] = 0 if j ≠ k. It is also shown in [4] that under certain conditions, a split Lie triple system with a coherent 0-root space is the direct sum of the family of its minimal ideals, each one being a simple split Lie triple system, and the simplicity of T is characterized. In the present paper we extend these results to arbitrary split Lie triple systems with no restrictions on their 0-root spaces.     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

Families of Finite Subsets of ℕ of Low Complexity and Tsirelson Type Spaces

We study Tsirelson type spaces of the form T[(ℳ︁_k, θ_k)^l_k=1] defined by a finite sequence (ℳ︁_k)^l_k=1 of compact families of finite subsets of ℕ. Using an appropriate index, denoted by i(ℳ︁), to measure the complexity of a family ℳ︁, we prove the following: If i(ℳ︁_k) < ω for all k = 1, …, l, then the space T[(ℳ︁_k, θ_k)^l_k=1] contains isomorphically some l_p, 1 < p < ∞, or c₀. If i(ℳ︁) = ω, then the space T[ℳ︁, θ] contains a subspace isomorphic to a subspace of the original Tsirelson's space.     

Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces

We prove that Lipschitz mappings are dense in the Newtonian–Sobolev classes N ^1,p (X, Y) of mappings from spaces X supporting p-Poincaré inequalities into a finite Lipschitz polyhedron Y if and only if Y is [p]-connected, π ₁(Y) = π ₂(Y) = · · · = π _[p](Y) = 0, where [p] is the largest integer less than or equal to p. This work was supported by the NSF grant DMS-0500966.     

On Strebel points

Any Beltrami coefficient μ on a hyperbolic Riemann surface X of infinite type represents a point [μ] _T in the Teichmüller space T(X) and a point [μ] _B in the tangent space of T(X) at the base point as well. The paper deals with the problem of determining whether that [μ] _T is a Strebel point is equivalent to that [μ] _B is an infinitesimal Strebel point.     

Hausdorff properties of topological algebras

Let P be a property of topological spaces. Let [P] be the class of all varieties having the property that any topological algebra in has underlying space satisfying property P. We show that if P is preserved by finite products, and if is preserved by ultraproducts, then [P] is a class of varieties that is definable by a Maltsev condition.?The property that all T ₀ topological algebras in are j-step Hausdor. (Hj) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to by showing that this topological implication holds in every (2j + 1)-permutable variety, but not in every (2j + 2)-permutable variety.?Finally, we show that the topological implication holds in every k-permutable, congruence modular variety. Received March 1, 2000; accepted in final form October 18, 2001.     

The set of indeterminate rings of a normal pair as a partially ordered set

Let R ì S{R\subset S} be an extension of integral domains and let [R, S] be the set of intermediate rings between R and S ordered by inclusion. If (R, S) is normal pair and [R, S] is finite, we do prove that there exists a semi-local Prüfer ring T with quotient field K such that [R,S] @ [T,K]{[R,S]\cong \lbrack T,K]} (as a partially ordered set). Consequently, any problem relative to the finiteness conditions in [R, S] can be investigated in the particular case where R is a semi-local Prüfer ring with quotient field S.     

Expansive Convergence Groups are Relatively Hyperbolic

Let a discrete group G act by homeomorphisms of a compactum in a way that the action is properly discontinuous on triples and cocompact on pairs. We prove that such an action is geometrically finite. The converse statement was proved by P. Tukia [T3]. So, we have another topological characterisation of geometrically finite convergence groups and, by the result of A. Yaman [Y2], of relatively hyperbolic groups. Further, if G is finitely generated then the parabolic subgroups are finitely generated and undistorted. This answer to a question of B. Bowditch and eliminates restrictions in some known theorems about relatively hyperbolic groups. Received: April 2007, Revision: May 2008, Accepted: August 2008     

A canonical structure theorem for finite joining-rank maps

A numerical isomorphism invariant,joining-rank, was introduced in [1] as a quantitative generalization of Rudolph’s property of minimal selfjoinings. Therein, a structure theory was developed for those transformationsT whose joining-rank, jr (T), is finite. Here, we sharpen the theorem and show it to be canonical: If jr (T)<∞ then there is a unique triple 〈e, p, S〉 wheree andp are natural numbers andS is a map with minimal self-joinings, such thatT is ane-point extension ofS ^P. Furthermore, the producte·p equals the joining-rank ofT. This theorem applies to any finite-rank mixing map, since for such maps the rank dominates the joining-rank. Another corollary is that any rank-1 transformation which is partial-mixing has minimal self-joinings. This partially answers a question of [3]. Partially supported by a National Science Foundation Postdoctoral Research Fellowship.     

On the equivalence of extremal Teichmüller mapping

Let T(Δ) and B(Δ) be the Teichmüller space and the infinitesimal Teichmüller space of the unit disk Δ respectively. In this paper, we show that [ν] _B(Δ) being an infinitesimal Strebel point does not imply that [ν] _T(Δ) is a Strebel point, vice versa. As an application of our results, problems proposed by Yao are solved. This work was supported by the National Natural Science Foundation of China (Grant No. 10571028)     

Group Gradings on Polynomial Algebras

Let k be a field. We consider gradings on a polynomial algebra k[X₁,…, X_n] by an arbitrary abelian group G, such that the indeterminates are homogeneous elements of nontrivial degree. We classify the isomorphism types of such gradings, and we count them in the case where G is finite. We present some examples of good gradings and find a minimal set of generators of the subalgebra of elements of trivial degree.     

Markov—Bernstein-Type Inequality for Trigonometric Polynomials with Respect to Doubling Weights on [-ω,ω]

   Abstract. Various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D. S. Lubinsky, we establish Markov—Bernstein-type inequalities for trigonometric polynomials with respect to doubling weights on [-ω,ω] . Namely, we show the theorem below. Theorem Let p ∈ [1,∞) and ω ∈ (0, 1/2] . Suppose W is a weight function on [-ω,ω] such that W(ω cos t) is a doubling weight. Then there is a constant C depending only on p and the doubling constant L so that

Free MVn-algebras

In this note we characterize free algebras in varieties of MV-algebras generated by a finite chain L _n as algebras of continuous functions from the spectrum of the Boolean skeleton of the free algebra into L _n . Received May 9, 2006; accepted in final form May 15, 2007.     

New spectra of strongly minimal theories in finite languages

We describe strongly minimal theories T_n with finite languages such that in the chain of countable models of T_n, only the first n models have recursive presentations. Also, we describe a strongly minimal theory with a finite language such that every non-saturated model has a recursive presentation.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

Norm Attaining Operators and Pseudospectrum

It is shown that if 1 < p < ∞ and X is a subspace or a quotient of an ℓ_p-direct sum of finite dimensional Banach spaces, then for any compact operator T on X such that ∥I + T∥ > 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥  > 1 and I + T does not attain its norm. The author would like to thank E. Shargorodsky for his interest and comments.