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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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References:
[1] G. Birkhoff: Lattice Theory. Providence, 1967. · Zbl 0153.02501
[2] R. Cignoli: Complete and atomic algebras of the infinite valued Łukasiewicz logic. Studia Logica 50 (1991), 375-384. · Zbl 0753.03026 · doi:10.1007/BF00370678
[3] C. J. Everett: Sequence completion of lattice moduls. Duke Math. J. 11 (1944), 109-119. · Zbl 0060.06301 · doi:10.1215/S0012-7094-44-01112-9
[4] D. Gluschankov: Cyclic ordered groups and \(MV\)-algebras. Czechoslovak Math. J. 43 (1993), 249-263.
[5] A. W. Hales: On the non-existence of free complete Boolean algebras. Fundam. Math. 54 (1964), 45-66. · Zbl 0119.26003 · eudml:213773
[6] J. Jakubík: Maximal Dedekind completion of an abelian lattice ordered group. Czechoslovak Math. J. 28 (1978), 611-631. · Zbl 0432.06012 · eudml:13091
[7] J. Jakubík: Direct product decompositions of \(MV\)-algebras. Czechoslovak Math. J. 44 (1994), 725-739. · Zbl 0821.06011 · eudml:31437
[8] J. Jakubík: On complete \(MV\)-algebras. Czechoslovak Math. J. 45 (1995), 473-480. · Zbl 0841.06010 · eudml:31482
[9] J. Jakubík: On archimedean \(MV\)-algebras. Czechoslovak Math. J 48 (1998), 575-582. · Zbl 0951.06011 · doi:10.1023/A:1022436113418 · eudml:30438
[10] M. Jakubíková: Über die \(B\)-Potenz einer teilweise geordneten Gruppe. Matem. časopis 23 (1973), 231-239. · Zbl 0277.06006 · eudml:30033
[11] M. Jakubíková: The nonexistence of free complete vector lattices. Časop. pěst. matem. 99 (1974), 142-146.
[12] D. Mundici: Interpretation of \(AFC^*\)-algebras in Łukasiewicz sentential calculus. Journ. Functional Anal. 65 (1986), 15-63. · Zbl 0597.46059 · doi:10.1016/0022-1236(86)90015-7
[13] R. Sikorski: Boolean Algebras. Second edition, Berlin, 1964. · Zbl 0123.01303
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