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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.)