Representations on Unigroups

  • David J. FoulisEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 111)


If G is an ordered abelian group with a generating order unit u, the order interval G+ [0, u]:={p ∈ G|0 ≤ pu} can be given the structure of an effect algebra in a natural way. Conversely, most effect algebras that arise in practice are interval effect algebras, i.e., are isomorphic to effect algebras of the form G+ [0, u]. The pair (G, u) is called a unigroup iff every abelian group-valued measure on the interval effect-algebra G+ [0, u] lifts to a (necessarily, unique) group homomorphism on G. An interval effect algebra has, up to isomophism, a unique representation as the order interval of a unigroup. Thus, a great part of the theory of effect algebras (and thus, of algebraic quantum logic) can be recast as a chapter of the theory of ordered abelian groups. In particular, the study of group actions on interval effect algebras amounts to the study of representations of groups on unigroups.


Abelian Group Unitary Representation Unit Interval Effect Algebra Positive Cone 
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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Massachusetts (Emeritus)AmherstUSA

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