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Complete generators and maximal completions of $$MV$$-algebras. (English) Zbl 0951.06010
An $$MV$$-algebra is an algebra with two binary, one unary, and two nullary operations satisfying a certain system of identities. By a suitable construction, it is possible to assign a distributive lattice $$L(A)$$ with a least and a greatest element to any $$MV$$-algebra $$A$$. An $$MV$$-algebra is called complete if the lattice $$L(A)$$ is complete. A subalgebra $$B$$ of an $$MV$$-algebra $$A$$ is said to be closed if $$L(B)$$ is a closed sublattice of $$L(A)$$. A subset $$X$$ of the carrier of a complete $$MV$$-algebra $$A$$ is referred to as a system of complete generators if any closed subalgebra of $$A$$ including $$X$$ coincides with $$A$$; if, moreover, any mapping of $$X$$ into a complete $$MV$$-algebra $$C$$ can be extended to a complete homomorphism of $$A$$ into $$C$$, the set $$X$$ is called a system of free complete generators; in such a case, $$A$$ is a referred to as a free complete $$MV$$-algebra with $$\alpha$$ free generators where $$\alpha =$$ card $$X$$.
The author proves the following theorem: If $$\beta$$ is an infinite cardinal, then there exists no free complete $$MV$$-algebra with $$\beta$$ free complete generators. The following results are demonstrated by examples: An $$MV$$-algebra with one generator need not be either complete or archimedean. There exist infinitely many nonisomorphic complete $$MV$$-algebras with one complete generator.
Furthermore, the author defines a so-called maximal completion of an $$MV$$-algebra and describes its construction.
Reviewer: M.Novotný (Brno)

##### MSC:
 06D35 MV-algebras 08B20 Free algebras
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