MAC NEILLE COMPLETION OF CENTERS AND CENTERS OF MAC NEILLE COMPLETIONS OF LATTICE EFFECT ALGEBRAS
If element z of a lattice effect algebra (E, circle plus, 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 circle plus. 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 (E) over cap of E which is its extension (i.e. E is densely embeddable into (E) over cap) then it is possible to embed E into a direct product of irreducible effect algebras. Thus E inherits some of the properties of (E) over cap. For example, the existence of a state in (E) over cap 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) over cap, i.e., if MC(C(E)) = C((E) over cap) 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.
center lattice effect algebra atom macneille completion