2,564 research outputs found
States on pseudo effect algebras and integrals
We show that every state on an interval pseudo effect algebra satisfying
some kind of the Riesz Decomposition Properties (RDP) is an integral through a
regular Borel probability measure defined on the Borel -algebra of a
Choquet simplex . In particular, if 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.
On a New Construction of Pseudo Effect Algebras
We define a new class of pseudo effect algebras, called kite pseudo effect
algebras, which is connected not necessarily with partially ordered groups, but
rather with generalized pseudo effect algebras where the greatest element is
not guaranteed. Starting even with a commutative generalized pseudo effect
algebra, we can obtain a non-commutative pseudo effect algebra. We show how
such kite pseudo effect algebras are tied with different types of the Riesz
Decomposition Properties. We find conditions when kite pseudo effect algebras
have the least non-trivial normal ideal.Comment: arXiv admin note: substantial text overlap with arXiv:1306.030
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
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 -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
Representation of States on Effect-Tribes and Effect Algebras by Integrals
We describe -additive states on effect-tribes by integrals.
Effect-tribes are monotone -complete effect algebras of functions where
operations are defined by points. Then we show that every state on an effect
algebra is an integral through a Borel regular probability measure. Finally, we
show that every -convex combination of extremal states on a monotone
-complete effect algebra is a Jauch-Piron state.Comment: 20 page
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
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