Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras

Abstract

We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element below x by x↓. For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x↓,x] is a subset of B. For every meager element (that means, an element x with x↓ = 0), the interval [0,x] is a complete MV-effect algebra. As a consequence, the set of all meager elements of an orthocomplete homogeneous effect algebra forms a commutative BCK-algebra with the relative cancellation property. We prove that a complete lattice ordered effect algebra E is completely determined by the complete orthomodular lattice S(E) of sharp elements, the BCK-algebra M(E) of meager elements and a mapping h:S(E)→2M(E) given by h(a) = [0,a] ∩ M(E).