Representations on Unigroups

Abstract

If G is an ordered abelian group with a generating order unit u, the order interval G⁺ [0, u]:={p ∈ G|0 ≤ p ≤ u} 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.