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Archimedean and block-finite lattice effect algebras. (English) Zbl 0967.06006
An effect algebra is a set \(E\) containing two distinguished elements 0, 1 and equipped with a partial commutative associative operation \(\oplus\) such that \(\forall a\in E\;\exists! b\in E: a\oplus b=1\) and \(1\oplus a\) is defined iff \(a=0\). A partial order \(\leq\) on \(E\) is defined by \(a\leq b\Leftrightarrow\exists c\in E: a\oplus c=b\). If \(E\) is a lattice with respect to this ordering, it is called a lattice effect algebra. From various definitions of a sub-effect algebra, \(Q\subseteq E\), the author uses the following: if \(a\oplus b=c\) and at least two elements of \(\{a,b,c\}\) are in \(Q\), then \(a,b,c\in Q\). According to previous results of the author, each lattice effect algebra is the union of its maximal compatible subsets (called blocks) which are MV-algebras. The question is when the MacNeille completion \(\widehat{E}\) of \(E\) (as a lattice) admits an extension of \(\oplus\) such that \(\widehat{E}\) becomes again an effect algebra. As a main result, a necessary and sufficient condition is given in the case of lattice effect algebras with finitely many blocks: The MacNeille completion of a lattice effect algebra \(E\) with finitely many blocks is a lattice effect algebra iff \(E\) is Archimedean, i.e., there is no \(e\in E\setminus\{0\}\) such that \(\bigoplus_{i=1}^n e\) exists for any \(n\in N\). Moreover, in this case the MacNeille completion of \(E\) may be obtained from the MacNeille completions of its blocks. The paper represents a clear and important contribution to this difficult topic.

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06D35 MV-algebras
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
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