Non-commutative effect algebras, L-algebras, and local duality

Abstract

GPE-algebras were introduced by Dvurečenskij and Vetterlein as unbounded pseudo-effect algebras. Recently, they have been characterized as partial L-algebras with local duality. In the present paper, GPE-algebras with an everywhere defined L-algebra operation are investigated. For example, linearly ordered GPE-algebra are of that type. They are characterized by their self-similar closures which are represented as negative cones of totally ordered groups. More generally, GPE-algebras with an everywhere defined multiplication are identified as negative cones of directed groups. If their partial L-algebra structure is globally defined, the enveloping group is lattice-ordered. For any self-similar L-algebra A, exponent maps are introduced, generalizing conjugation in the structure group. It is proved that the exponent maps are L-algebra automorphisms of A if and only if A is a GPE-algebra. As an application, a new characterization of cone algebras is obtained. Lattice GPE-algebras are shown to be equivalent to ∧-closed L-algebras with local duality.