Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements

We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra $E$ is separable and modular then there exists a faithful state on $E$. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra $\widehat{E}$ and the compatiblity center of $E$ is not a Boolean algebra then there exists an $(o)$-continuous subadditive state on $E$

Sharply Orthocomplete Effect Algebras

Special types of effect algebras $E$ called sharply dominating and S-dominating were introduced by S. Gudder in \cite{gudder1,gudder2}. We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of $E$. Namely we prove that in every sharply orthocomplete S-dominating effect algebra $E$ the set of sharp elements and the center of $E$ are complete lattices bifull in $E$. If an Archimedean atomic lattice effect algebra $E$ is sharply orthocomplete then it is complete

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

Two-valued states on Baer ∗^*-semigroups

In this paper we develop an algebraic framework that allows us to extend families of two-valued states on orthomodular lattices to Baer $^*$-semigroups. We apply this general approach to study the full class of two-valued states and the subclass of Jauch-Piron two-valued states on Baer $^*$-semigroups.Comment: Reports on mathematical physics (accepted 2013