Computable Families of Superatomic Boolean Algebras

We describe the families of superatomic Boolean algebras which have a computable numbering. We define the notion of majorizability and establish a criterion that is formulated only on using algorithmic terms and majorizability. We give some examples showing that the condition of majorizability is essential. We also prove some criterion for the existence of a computable numbering for a family of -atomic algebras ( is a computable ordinal).     

Degree Spectra of Relations on Boolean Algebras

We show that every computable relation on a computable Boolean algebra is either definable by a quantifier-free formula with constants from (in which case it is obviously intrinsically computable) or has infinite degree spectrum.     

Iterative Algebras without Projections

We deal with iterative algebras of functions of -valued logic lacking projections, which we call algebras without projections. It is shown that a partially ordered set of algebras of functions of -valued logic, for , without projections contains an interval isomorphic to the lattice of all iterative algebras of functions of -valued logic. It is found out that every algebra without projections is contained in some maximal algebra without projections, which is the stabilizer of a semigroup of non-surjective transformations of the basic set. It is proved that the stabilizer of a semigroup of all monotone non-surjective transformations of a linearly ordered 3-element set is not a maximal algebra without projections, but the stabilizer of a semigroup of all transformations preserving an arbitrary non one-element subset of the basic set is.     

Recursive Homogeneous Boolean Algebras

Isomorphism types of countable homogeneous Boolean algebras are described in [1], in which too is settled the question of whether such algebras are decidable. Precisely, a countable homogeneous Boolean algebra has a decidable presentation iff the set by which an isomorphism type of that algebra is characterized belongs to a class of the arithmetic hierarchy. The problem of obtaining a characterization for homogeneous Boolean algebras which have a recursive presentation remained open. Partially, here we resolve this problem, viz., estimate an exact upper and an exact lower bounds for the set which an isomorphism type of such any algebra is characterized by in terms of the Feiner hierarchy.     

On Initial Segments of Computable Linear Orders

We show there is a computable linear order with a initial segment that is not isomorphic to any computable linear order.     

Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces

We introduce the notion of the -prolongation of Lie algebras of differential operators on homogeneous spaces. The -prolongations are topological invariants that coincide with one-dimensional cohomologies of the corresponding Lie algebras in the case where V is a homogeneous space. We apply the obtained results to the spaces S ¹ (the Virasoro algebra) and .     

Relatively Hyperimmune Relations on Structures

Let be a computable structure and let R be an additional relation on its domain. We establish a necessary and sufficient condition for the existence of an isomorphic copy of such that the image of R is h-simple (h-immune) relative to .     

Complexity of some natural problems on the class of computable I-algebras

We study computable Boolean algebras with distinguished ideals (I-algebras for short). We prove that the isomorphism problem for computable I-algebras is Σ ₁ ¹ -complete and show that the computable isomorphism problem and the computable categoricity problem for computable I-algebras are Σ ₃ ⁰ -complete.     

Lattice Ordered Polynomial Algebras

In this paper we define a lattice order on a set F of binary functions. We then provide necessary and sufficient conditions for the resulting algebra F to be a distributive lattice or a Boolean algebra. We also prove a Cayley theorem for distributive lattices by showing that for every distributive lattice , there is an algebra F of binary functions, such that is isomorphic to F and we show that F is a distributive lattice iff the operations and are idempotent and cummutative, showing that this result cannot be generalized to non-distributive lattices or quasilattices without changing the definitions of and . We also examine the equational properties of an Algebra for which , now defined on the set of binary -polynomials is a lattice or Boolean algebra.     

Corona Extendibility and Asymptotic Multiplicativity

We study outer multiplier algebras, C(E)=M(E)/E, also known as corona algebras, and *-homomorphisms A C(E) . We prove in several instances that for all such maps there must exist an extension to a largerC ^* -algebra . The Kasparov Technical Theorem gives one class of examples where . Our theorems apply to subhomogeneous C ^* -algebras, such as , the algebra used in Cuntz's picture of K-theory. Where such an extension theorem exists, there must exist an asymptotic morphism whose restriction to A is equivalent to the identity. We also use extension results to prove closure properties for the collection of C ^*-algebras that have stable relations.     

A note on normal varieties of monounary algebras

A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. Let L denote the lattice of all varieties of monounary algebras (A,f) and let V be a non-trivial non-normal element of L. Then V is of the form with some n > 0. It is shown that the smallest normal variety containing V is contained in for every m > 1 where C denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of L consisting of all normal elements of L is isomorphic to L.     

Universal categories of uniform and metric locales

Recall that a category is called universal if it contains an isomorphic copy of any concrete category as a full subcategory. In particular, if is universal then every monoid can be represented as the endomorphism monoid of an object in . A major obstacle to universality in categories of topological nature are the constant maps (which prevent, for instance, representing nontrivial groups as endomorphism monoids). Thus, to obtain, say, a universal category of uniform spaces, the constants have to be prohibited by artificial additional conditions (for instance, conditions of an openness type). Since in generalized spaces (locales) we do not necessarily have points, the question naturally arises as to whether we can get rid of surplus conditions in search of universality there. In this paper we prove that the category of uniform locales with all uniform morphisms is universal. Indeed we establish the universality already for the subcategory of very special uniform locales, namely Boolean metric ones. Moreover, universality is also obtained for more general morphisms, such as Cauchy morphisms, as well as for special metric choices of morphisms (contractive, Lipschitz). The question whether one can avoid uniformities remains in general open: we do not know whether the category of all locales with all localic morphisms is universal. However, the answer is final for the Boolean case: by a result of McKenzie and Monk ([10], see Section 4) one cannot represent groups by endomorphisms of Boolean algebras without restriction by an additional structure.We use only basic categorical terminology, say, that from the introductory chapters of [9]. All the necesasary facts concerning generalized spaces (frames, locales) and universality are explicitly stated. More detail on frames (locales) can be found in [8] and on universality and embeddings of categories in [11].Presented by E. Fried.     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

On the triple jump of the set of atoms of a Boolean algebra

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let be a computable Boolean algebra with infinitely many atoms and be the Turing degree of the atom relation of . If is a c.e. degree such that , then there is a computable copy of where the atom relation has degree . In particular, for every c.e. degree , any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree .

     

Algebras between C *(X) and C(X) that are closed under countable composition

Let be the algebra of all real-valued continuous functions on a completely regular space X, and the subalgebra of bounded functions. We show that the space of real maximal ideals of an intermediate algebra between and is a -embedded closed subspace of the product of with a product of copies of the real line. We make use of this embedding to provide a new characterization of the intermediate algebras that are closed under countable composition, and to exhibit an example of an intermediate algebra on that is closed under countable composition but not isomorphic to any .     

The space of minimal prime ideals in a 0-distributive semilattice

LetS be a 0-distributive semilattice and be its minimal spectrum. It is shown that is Hausdorff. The compactness of has been characterized in several ways. A representation theorem (like Stone's theorem for Boolean algebras) for disjunctive, 0-distributive semilattices is obtained.     

The ideal envelope of an operator algebra

A left ideal of any -algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here that operator algebras with a r.c.a.i. should be studied in terms of a certain left ideal of a -algebra. We study operator algebras and their multiplier algebras from the perspective of ``Hamana theory' and using the multiplier algebras introduced by the first author.