Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras

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 ([18]). This means that if there exists a MacNeille completion Ê of E which is its extension (i.e. E is densely embeddable into Ê) then it is possible to embed E into a direct product of irreducible effect algebras. Thus E inherits some of the properties of Ê. For example, the existence of a state in Ê 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 Ê, i.e., if MC(C(E)) = C(Ê) 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.