Downward closure of depth in countable Boolean algebras

We study the set of depths of relative algebras of countable Boolean algebras, in particular the extent to which this set may not be downward closed within the countable ordinals for a fixed countable Boolean algebra. Doing so, we exhibit a structural difference between the class of arbitrary rank countable Boolean algebras and the class of rank one countable Boolean algebras.     

Initial Chain Algebras on Pseudotrees

We investigate a new class of Boolean algebra, called initial chain algebras on pseudotrees. We discuss the relationship between this class and other classes of Boolean algebras. Every interval algebra, and hence every countable Boolean algebra, is an initial chain algebra. Every initial chain algebra on a tree is a superatomic Boolean algebra, and every initial chain algebra on a pseudotree is a minimally-generated Boolean algebra.We show that a free product of two infinite Boolean algebras is an initial chain algebra if and only if both factors are countable.     

Composition algebras of the second kind

The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in a countable dimension. The constructed algebras each contains a non-isotropic idempotent e² = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A, the group Ortaut A of orthogonal automorphisms is specified. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 428–447, July–August, 2007.     

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

     

Lower bounds for algebraic complexity of nilpotent associative algebras

Exact algebraic algorithms for calculating the product of two elements of nilpotent associative algebras over fields of characteristic zero are considered (this is a particular case of simultaneous calculation of several multinomials). The complexity of an algebra in this computational model is defined as the number of nonscalar multiplications of an optimal algorithm. Lower bounds for the tensor rank of nilpotent associative algebras (in terms of dimensions of certain subalgebras) are obtained, which give lower bounds for the algebraic complexity of this class of algebras. Examples of reaching these estimates for different dimensions of nilpotent algebras are presented.     

Constructibility of Boolean algebras of elementary characteristic (1,0,1)

We deal with problems of finding a criterion of being strongly constructivizable for Boolean algebras. An example of a constructive but not strongly constructivizable Boolean algebra of characteristic (1, 0, 1) with a decidable set of atoms is constructed, and the construction is then generalized to the case of an arbitrary characteristic (k+1,0,1). Supported by the RFFR grant No. 96-01-01525. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 499–521, September–October, 1998.     

Metric Boolean algebras and constructive measure theory

 This work concerns constructive aspects of measure theory. By considering metric completions of Boolean algebras – an approach first suggested by Kolmogorov – one can give a very simple construction of e.g. the Lebesgue measure on the unit interval. The integration spaces of Bishop and Cheng turn out to give examples of such Boolean algebras. We analyse next the notion of Borel subsets. We show that the algebra of such subsets can be characterised in a pointfree and constructive way by an initiality condition. We then use our work to define in a purely inductive way the measure of Borel subsets. Received: 9 November 2000 / Revised version: 23 March 2001 / Published online: 12 July 2002     

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.     

Conglomerability of probability measures on Boolean algebras

For the case where is a Boolean algebra of events and P is a probability (finitely additive) [23.] considered the question of conglomerability of P and found that in many circumstances this natural notion was equivalent to countable additivity of P. [32.] pursued these investigations on the connection between countable additivity and conglomerability in greater detail for the case where is a σ-algebra. [29.] and [37.] give alternative proofs. This article is an extension (for the most part) of Schervish, Seidenfeld, and Kadane's work to the case where is an arbitrary Boolean algebra. The more restrictive notion of positive conglomerability for a class of algebras, including the countable algebras, σ-complete algebras, and inifinite product algebras is treated completely. This class is described by the requirement that a {0, 1}-valued measure be countably additive if every countable family of negligible sets is contained within a negligible set (i.e., corresponds to a P-point of of the Stone space). In general positive conglomerability fails to be equivalent to countable additivity though the degree of failure is minor. Building on techniques of Hill, Lane, and Zame, we obtain partial results on conglomerability for non-σ-complete algebras.     

Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220.     

Implicative twist-structures

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.     

Definability of Boolean Algebras in \mathbb{H}\mathbb{F} -Superstructures

Within the frames of the -definability approach propounded by Yu. L. Ershov, we study into the definability of Boolean algebras and their Frechet ranks in hereditarily finite superstructures. Examples are constructed of a superatomic Boolean algebra whose Frechet rank is not -definable in the hereditarily finite superstructure over that algebra, and of an admissible set in which the atomless Boolean algebra is not autostable.     

BOOLEAN ALGEBRAS IN AST

In this paper we investigate Boolean algebras and their subalgebras in Alternative Set Theory (AST). We show that any two countable atomless Boolean algebras are isomorphic and we give an example of such a Boolean algebra. One other main result is, that there is an infinite Boolean algebra freely generated by a set. At the end of the paper we show that the sentence “There is no non-trivial free group which is a set” is consistent with AST.     

Locally finite dimensional lie algebras and their derivation algebras

The derivation algebras of all locally finite dimensional locally simple Lie algebras over a field of characteristic 0 are determined. Every locally finite dimensional Lie algebra of countable dimension is a subalgebra of the outer derivation algebra outder (ℒ) for every Lie algebra ℒ, which is the direct limit of diagonally embedded classical Lie algebras. These outer derivation algebras have dimension ℒ and are never locally finite dimensional. Dedicated to Prof. H. Petersson on the occasion of his 60th birthday     

On computation complexity problems concerning relation algebras

In this paper, we study the computation complexity of some algebraic combinatorial problems that are closely associated with the graph isomorphism problem. The key point of our considerations is a relation algebra which is a combinatorial analog of a cellular algebra. We present upper bounds on the complexity of central algorithms for relation algebras such as finding the standard basis of extensions and intersection of relation algebras. Also, an approach is described to the graph isomorphism problem involving Schurian relation algebras (these algebras arise from the centralizing rings of permutation groups). We also discuss a number of open problems and possible directions for further investigation. Bibliography: 18 titles. Translated by I. N.Ponomarenko. Translated fromZapiski Nauchnykh Seminarov POMI, Vol 202, 1992, pp. 116–134.     

Orthogonal modules and nonlinear cohomologies

Ordinary cohomologies in an algebra most often exhibit themselves either as objects that describe extensions (of algebras, groups, or modules) or as subjects hindering the existence of algebraic constructions. We introduce the notion of nonlinear cohomologies at a level of cochain complexes, furnish particular examples for Lie algebras, and give an interpretation of first nonlinear cohomologies in terms of “quasiextensions” of orthogonal modules. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 522–541, September–October, 1998.     

Graded Lie algebras whose Cartan subalgebra is the algebra of polynomials in one variable

We define a class of infinite-dimensional Lie algebras that generalize the universal enveloping algebra of the algebra sl(2, ℂ) regarded as a Lie algebra. These algebras are a special case of ℤ-graded Lie algebras with a continuous root system, namely, their Cartan subalgebra is the algebra of polynomials in one variable. The continuous limit of these algebras defines new Poisson brackets on algebraic surfaces. In memory of M. V. Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 345–352, May, 2000.     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

Bases, filtrations and module decompositions of free Lie algebras

We use the technique known as elimination to devise some new bases of the free Lie algebra which (like classical Hall bases) consist of Lie products of left normed basic Lie monomials. Our bases yield direct decompositions of the homogeneous components of the free Lie algebra with direct summands that are particularly easy to describe: they are tensor products of metabelian Lie powers. They also give rise to new filtrations and decompositions of free Lie algebras as modules for groups of graded algebra automorphisms. In particular, we obtain some new decompositions for free Lie algebras and free restricted Lie algebras over fields of positive characteristic.     

Fuzzy Boolean algebras

The purpose of this paper is to generalize the following situation: from the concrete structure $\text{B}$, we define the notion of Boolean algebras; the Stone representation theorem allows us to replace the algebraic study of Boolean algebras by a topological one. Let E be a non-empty set, and J a non-empty ordered set. Note $\text{B}$ the set of all fuzzy subsets of (E,J). We shall introduce the concept of fuzzy Boolean algebra and find a representation theorem. But it will be difficult to speak of the dual fuzzy topological space of a fuzzy Boolean algebra as we shall see further, except in certain particular cases.