Enumerated algebras with uniformly recursive-separable classes

Tashkent. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 5, pp. 85–102, September–October, 1993.     

Restricted Lie algebras with bounded cohomology and related classes of algebras

We determine the structure of restricted Lie algebras with bounded cohomology over arbitrary fields of prime characteristic. As a byproduct a classification of the serial restricted Lie algebras and the restricted Lie algebras of finite representation type is obtained. In addition, we derive complete information on the finite dimensional indecomposable restricted modules of these algebras over algebraically closed fields.     

Sets with B-action and linear algebra

Let B be a Boolean ring. Any unital B-module M has a B-action given by for all . We show that any set with B-action may be represented as a subset with B-action of a unital B-module. We extend this to algebras with B-action and apply it to if-then-else algebras over Boolean algebras by viewing them as semilattices with B-action. Received May 23, 1996; accepted in final form February 4, 1998.     

Natural classes of universal algebras

A natural class of universal algebras is a class which is closed under isomorphism, subalgebras, disjoint suprema, and essential extensions. In suitable varieties, natural classes form a boolean lattice, and lead to a decomposition of any universal algebra into continuous molecular, discrete, and bottomless subalgebras.Dedicated to Professor John DaunsReceived March 29, 2004; accepted in final form May 20, 2004     

Positive equivalences with finite classes and related algebras

Tashkent. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 196–200, September–October, 1992.     

Chem classes and Lie-Rinehart algebras

Let A be a F-algebra where F is a field, and let W be an A-module of finite presentation. We use the linear Lie-Rinehart algebra V_W of W to define the first Chern-class c₁(W) in , where U in Spec(A) is the open subset where W is locally free. We compute explicitly algebraic V_W-connections on maximal Cohen-Macaulay modules W on the hypersurface-singularities B_mn2 = x^m + yⁿ + z², and show that these connections are integrable, hence the first Chern-class c₁(W) vanishes. We also look at indecomposable maximal Cohen-Macaulay modules on quotient-singularities in dimension 2, and prove that their first Chern-class vanish.     

Some classes of pre-Hilbert algebras with norm-one central idempotent

In this paper we first characterize the pre-Hilbert algebras with a norm-one central idempotent e such that ‖ex‖ = ‖x‖ for any x ∈ A. This generalizes a well-known theorem by Ingelstam asserting that every alternative pre-Hilbert algebra with a unit 1 such that ‖1‖ = 1 is isomorphic to ?, ?, ? or $\mathbb{O}$ . We also show that every power-associative pre-Hilbert algebra satisfying ‖x ²‖ = ‖x‖² for every element has a unique nonzero idempotent, which is a unit element. In fact, the same conclusion will be proved in a more general setting. As application we give some conditions characterizing when a real algebra A, which is a prehilbert space, is isomorphic to one of the Hilbert algebras ?, ?, ? or $\mathbb{O}$ .     

On direct limit classes of algebras

We investigate classes of algebras which can be obtained by a direct limit construction from an algebra. We generalize some results from monounary algebras. Supported by grant VEGA 2/5065/5.     

Torsion classes of generalized Boolean algebras

Torsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T _g and R _g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T _g and R _g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T _g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized Boolean algebras.