Relation algebras and function semigroups

The concept of relation algebra unifies many familiar notions from algebra (especially those of systems having “natural” models as groups, Boolean algebras etc.). The fundamental theorem on relation algebras asserts the existence of simple conditions which characterize any given class of relation algebras. The idea of relation algebra is very useful for the study of function and transformation semigroups, which is the central part of the theory of semigroups. It provides a general outlook, permits one to formulate many natural problems, and ensures that these problems possess non-trivial solutions. A number of examples illustrate this point.     

Small semigroups in topological algebras

The research was partially supported by the International Science Foundation.     

Continuous semigroups in locally convex algebras

Continuity conditions of one-parametric semigroups in locally convex algebras of general form are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 154–158, February, 1991.     

Generalized Mehler semigroups and applications

Summary We construct and study generalized Mehler semigroups (p _t ) _t 0 and their associated Markov processesM. The construction methods for (p _t ) _t 0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert spaceH can be extended to some larger Hilbert spaceE, with the embeddingHE being Hilbert-Schmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedX _t =C dW _t +AX _t dt (with possibly unbounded linear operatorsA andC onH) on a suitably chosen larger spaceE. For Gaussian generalized Mehler semigroups (p _t ) _t 0 with corresponding Markov processM, the associated (non-symmetric) Dirichlet forms (E D(E)) are explicitly calculated and a necessary and sufficient condition for path regularity ofM in terms of (E,D(E)) is proved. Then, using Dirichlet form methods it is shown thatM weakly solves the above stochastic differential equation if the state spaceE is chosen appropriately. Finally, we discuss the differences between these two methods yielding strong resp. weak solutions.     

Prime and Semiprime Semigroup algebras of cancellative semigroups

Necessary and sufficient conditions are given for a semigroup algebra of a cancellative semigroup to be prime and semiprime. These conditions were proved necessary by Okninski; our contribution is to show that they are also sufficient. The techniques used in the proof are a new variation on the -methods which were developed originally for group algebras.

     

Representations of finite universal algebras in finite semigroups

An answer is given to the question of what conditions must be imposed on a signature in order that every finite algebra with this signature can be represented in the Mal'tsev-Cohn sense in a finite semigroup.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 285–290, March, 1971.The author wishes to thank A. G. Kurosh and Yu. K. Rebane for directing this work.     

Generalized symmetrization in enveloping algebras

Let g be a Lie algebra,S(g) the symmetric algebra,U(g) the universal enveloping algebra, andZ(g) the center ofU(g). The aim of this paper is to discuss a construction of a class of linear isomorphisms :S(g)U(g) which commute with the adjoint representation. Applications to constructing a basis inZ(g) for classical g are also sketched.Supported by the Russian Foundation for Basic Research under grant 95-01-00814     

Generalized projections in Banach algebras

In this note we investigate generalized projections in Banach algebras. Our results generalize results obtained for bounded linear operators on Hilbert spaces.     

Finite degree: algebras in general and semigroups in particular

An algebra A has finite degree if its term functions are determined by some finite set of finitary relations on A. We study this concept for finite algebras in general and for finite semigroups in particular. For example, we show that every finite nilpotent semigroup has finite degree (more generally, every finite algebra with bounded p _n -sequence), and every finite commutative semigroup has finite degree. We give an example of a five-element unary semigroup that has infinite degree. We also give examples to show that finite degree is not preserved in general under taking subalgebras, homomorphic images, direct products or subdirect factors.     

Generalized Clifford algebras and certain infinite dimensional Lie algebras

The embedding of certain infinite dimensional Lie algebras in generalized Clifford algobras C(N, p) is given. The correspondence between C(N,2) andgl(N, C) asN⇌∞ is pointed out.     

Generalized state machines and generalized transformation semigroups

We introduce the concepts of a generalized state machine and a generalized transformation semigroup and discuss their algebraic properties.     

Realizing irreducible semigroups and real algebras of compact operators

Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X) as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real?