Zdenka Riečanová

Generalization of Blocks for D-Lattices and Lattice- Ordered Effect Algebras

We show that everyD-lattice (lattice-ordered effect algebra)P is a set-theoreticunion of maximal subsets of mutually compatible elements, called blocks.Moreover, blocks are sub-D-lattices and sub-effect-algebras ofP which areMV-algebras closed with respect to all suprema and infima existing inP.

Continuous lattice effect algebras admitting order-continuous states

We prove that a necessary and sufficient condition for the existence of a faithful ((o)-continuous) state on a complete modular atomic effect algebra E is the separability of E . Moreover, we generalize the famous Kaplansky theorem about order continuity of complemented complete modular lattices onto complete modular atomic effect algebras. Some other statements about the algebraic structure of modular effect algebras are shown. We prove that every chain and every block in an irreducible complete modular atomic effect algebra is finite. Moreover, every complete atomic modular effect algebra is compactly generated by finite elements.

Smearings of States Defined on Sharp Elements Onto Effect Algebras

In some sense, a lattice effect algebra E is a smeared orthomodular lattice S(E), which then becomes the set of all sharp elements of the effect algebra E. We show that if E is complete, atomic, and (o)-continuous, then a state on E exists iff there exists a state on S(E). Further, it is shown that such an effect algebra E is an algebraic lattice compactly generated by finite elements of E. Moreover, every element of E has a unique basic decomposition into a sum of a sharp element and a ⊕-orthogonal set of unsharp multiples of atoms.

Proper Effect Algebras Admitting No States

We show that there is even a finite proper effect algebra admitting no states. Further, every lattice effect algebra with an ordering set of valuations is an MV effect algebra (consequently it can be organized into an MV algebra). An example of a regular effect algebra admitting no ordering set of states is given. We prove that an Archimedean atomic lattice effect algebra is an MV effect algebra iff it admits an ordering set of valuations. Finally we show that every nonmodular complete effect algebra with trivial center admits no order-continuous valuations.

ORTHOGONAL SETS IN EFFECT ALGEBRAS

We show that for a lattice effect algebra two conceptions of completeness (cr-completeness) coincide. Moreover, a separable effect algebra is complete if and only if it is cr-complete. Further, in an Archimedean atomic lattice effect algebra to every nonzero element x there is a ©-orthogonal system G of not necessary different atoms such that x =

EFFECT ALGEBRAS OF POSITIVE LINEAR OPERATORS DENSELY DEFINED ON HILBERT SPACES

We show that the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space can be equipped with partial sum of operators making it a generalized effect algebra. This sum coincides with the usual sum of two operators whenever it exists. Moreover, blocks of this generalized effect algebra are proper sub-generalized effect algebras. All intervals in this generalized effect algebra become effect algebras which are Archimedean, convex, interval effect algebras, for which the set of vector states is order determining. Further, these interval operator effect algebras possess faithful states.

States on sharply dominating effect algebras

We prove that sharply dominating Archimedean atomic lattice effect algebras can be characterized by the property called basic decomposition of elements. As an application we prove the state smearing theorem for these effect algebras.

Subdirect Decompositions of Lattice Effect Algebras

We prove a theorem about subdirect decompositions of lattice effect algebras. Further, we show how, under these decompositions, blocks, sets of sharp elements and centers of those effects algebras are decomposed. As an application we prove a statement about the existence of subadditive state on some block-finite effect algebras.

Lattice effect algebras with (o)-continuous faithful valuations☆

We prove that if there exists an order-continuous, faithful valuation ω on a lattice effect algebra E then E is modular, separable and order-continuous. It is also shown that such an effect algebra E can be supremum and infimum densely embedded into a complete effect algebra E which is also modular separable and order-continuous, since the valuation ω can be extended to a unique order-continuous faithful valuation ω on E.

States, uniformities and metrics on lattice effect algebras

We show that every state ω on a lattice effect algebra E induces a uniform topology on E. If ω is subadditive this topology coincides with pseudometric topology induced by ω. Further, we show relations between the interval and order topology on E and topologies induced by states.