Does there exist an atomic lattice effect algebra with non-atomic subalgebra of sharp elements? An affirmative answer to this question (and slightly more) is given: An example of an atomic MV-effect algebra with a non-atomic Boolean subalgebra of sharp or central elements is presented.
We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.
If element z of a lattice effect algebra (E,⊕,0,1) is central, then the interval [0,z] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕. It is a known fact that if the set C(E) of central elements of E is an atomic Boolean algebra and the supremum of all atoms of C(E) in E equals to the top element of E, then E is isomorphic to a subdirect product of irreducible effect algebras (\cite{R2}). This means that if there exists a MacNeille completion E^ of E which is its extension (i.e. E is densely embeddable into E^) then it is possible to embed E into a direct product of irreducible effect algebras. Thus E inherits some of the properties of E^. For example, the existence of a state in E^ implies the existence of a state in E. In this context, a natural question arises if the MacNeille completion of the center of E (denoted as MC(C(E))) is necessarily the same as the center of E^, i.e., if MC(C(E))=C(E^) is necessarily true. We show that the equality is not necessarily fulfilled. We find a necessary condition under which the equality may hold. Moreover, we show also that even the completeness of C(E) and its bifullness in E is not sufficient to guarantee the mentioned equality.
Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered σ-effect algebras and their "smearings" with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a σ-MV algebra, and every observable is defined by a smearing of a sharp observable, which is obtained from generalized Loomis-Sikorski theorem for σ-MV algebras. Generalized observables with the range in the set of sharp real observables are studied and it is shown that they contain all smearings of observables.
If element z of a lattice effect algebra (E,⊕,0,1) is central, then the interval [0,z] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕. It is a known fact that if the set C(E) of central elements of E is an atomic Boolean algebra and the supremum of all atoms of C(E) in E equals to the top element of E, then E is isomorphic to a direct product of irreducible effect algebras (\cite{R2}). In \cite{PR} Paseka and Riečanová published as open problem whether C(E) is a bifull sublattice of an Archimedean atomic lattice effect algebra E. We show that there exists a lattice effect algebra (E,⊕,0,1) with atomic C(E) which is not a bifull sublattice of E. Moreover, we show that also B(E), the center of compatibility, may not be a bifull sublattice of E.