Decompositions of Measures on Pseudo Effect Algebras

Recently in \cite{Dvu3} it was shown that if a pseudo effect algebra satisfies a kind of the Riesz Decomposition Property ((RDP) for short), then its state space is either empty or a nonempty simplex. This will allow us to prove a Yosida-Hewitt type and a Lebesgue type decomposition for measures on pseudo effect algebra with (RDP). The simplex structure of the state space will entail not only the existence of such a decomposition but also its uniqueness

Type-Decomposition of a Pseudo-Effect Algebra

The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.Comment: 18 page

The Lattice and Simplex Structure of States on Pseudo Effect Algebras

We study states, measures, and signed measures on pseudo effect algebras with some kind of the Riesz Decomposition Property, (RDP). We show that the set of all Jordan signed measures is always an Abelian Dedekind complete $\ell$-group. Therefore, the state space of the pseudo effect algebra with (RDP) is either empty or a nonempty Choquet simplex or even a Bauer simplex. This will allow represent states on pseudo effect algebras by standard integrals

Kite Pseudo Effect Algebras

We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which is connected with partially ordered groups not necessarily with strong unit. In such a case, starting even with an Abelian po-group, we can obtain a noncommutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz Decomposition Properties. Kites are so-called perfect pseudo effect algebras, and we define conditions when kite pseudo effect algebras have the least non-trivial normal ideal

Lexicographic Effect Algebras

In the paper we investigate a class of effect algebras which can be represented in the form of the lexicographic product \Gamma(H\lex G,(u,0)), where $(H,u)$ is an Abelian unital po-group and $G$ is an Abelian directed po-group. We study algebraic conditions when an effect algebra is of this form. Fixing a unital po-group $(H,u)$, the category of strong $(H,u)$-perfect effect algebra is introduced and it is shown that it is categorically equivalent to the category of directed po-group with interpolation. We show some representation theorems including a subdirect product representation by antilattice lexicographic effect algebras

States on pseudo effect algebras and integrals

We show that every state on an interval pseudo effect algebra $E$ satisfying some kind of the Riesz Decomposition Properties (RDP) is an integral through a regular Borel probability measure defined on the Borel $\sigma$-algebra of a Choquet simplex $K$. In particular, if $E$ satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of $K.