Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras

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 BCKalgebra 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).

• Type: Journal
• Year: 2010
• Where: ORDER
• Authors: Gejza Jenca
• Keywords: Algebra, Meager Elements, ORDER 2010, Orthocomplete Homogeneous Effect