Lattice effect algebras densely embeddable into complete ones

An effect algebraic partial binary operation ⊕ defined on the underlying set E uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion E of E there exists an effect algebraic partial binary operation ⊕ then ⊕ need not be an extension of ⊕. Moreover, for an Archimedean atomic lattice effect algebra E we give a necessary and sufficient condition for that ⊕ existing on E is an extension of ⊕ defined on E. Further we show that such ⊕ extending ⊕ exists at most one.

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