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Sharp and meager elements in orthocomplete homogeneous effect algebras. (English) Zbl 1193.03084
According to [G. Jenča, Bull. Aust. Math. Soc. 64, No. 1, 81–98 (2001; Zbl 0985.03063)], a homogeneous effect algebra is one in which \(u \leq v_1 \oplus v_2 \leq u'\) implies that \(u = u_1 \oplus u_2\) for some \(u_1 \leq v_1\) and \(u_2 \leq v_2\). An element \(x\) of an effect algebra \(E\) is said to be sharp if \(x \wedge x'= 0\), and meager if \(0\) is the single sharp element below \(x\). If \(E\) is lattice-ordered and complete, then the set \(S(E)\) of all its sharp elements is known to be a complete sublattice. The author shows that if \(E\) is orthocomplete and homogeneous, then the subset \(M(E)\) of its meager elements forms a commutative BCK-algebra with the relative cancellation property.
The main result of the paper states that a complete lattice effect algebra \(E\) is, up to isomorphism, characterised by the triple \((S(E), M(E), h)\), where \(h\) is the mapping that associates the lower end \(\{x \in M(E): x \leq a\}\) of \(M(E)\) to every \(a \in S(E)\). The proof depends on the fact that \(E\) is actually orthocomplete and homogeneous. An explicit construction of a copy of \(E\), resemblig the triple construction of Stone algebras, is presented. The inverse problem – which triples consisting of a lattice effect algebra \(S\), BCK-algebra \(M\) and an appropriate function \(h\) arise from a complete lattice effect algebra – is not addressed in the paper.

03G12 Quantum logic
06C15 Complemented lattices, orthocomplemented lattices and posets
06F35 BCK-algebras, BCI-algebras
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
Zbl 0985.03063
Full Text: DOI
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