Quantum structures and partial algebras
by
Thomas Vetterlein
Mathematical Institute, Slovak Academy of Sciences, Stefanikova 49, 81473 Bratislava, Slovakia

Quantum structures are assumed to be algebraic structures which are in some way related to the formalism of quantum physics. Quite a number of partial algebras which are considered to belong to this class has been studied during the last decades. Orthomodular algebras (OMAs), also known as orthomodular posets, have probably been examined most intensively. They are modeled upon the lattice of closed subspaces of a Hilbert space; and the exact conditions were found for an OMA to be representable as such a lattice.

Not long ago, the more general effect algebras have been introduced; they are axiomatized in a way similar to OMAs, but one axiom is dropped. Since their physical prototype is the interval of a po-group, conditions have been investigated under which an effect algebra is representable by a po-group, and certain sufficient criteria were found.

We are concerned with a further generalization of effect algebras: with pseudoeffect (PE-)algebras, which, in contrast to effect algebras, are not necessarily commutative. We shall see that quite a number of results concerning the structure of effect algebras may be generalized to the non-commutative case. In particular, PE-algebras fulfilling a certain kind of Riesz property are representable as a po-group interval.

Date received: January 23, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caht-10.