zbMATH — the first resource for mathematics

Regular representations of semisimple MV-algebras by continuous real functions. (English) Zbl 0987.06010
With any topological space \(M\) one can associate the MV-algebra \(A_1(M)\) corresponding to the \(\ell \)-group of all continuous bounded real functions on \(M\). If \(A\) is an MV-algebra and \(M\) a topological space, then an injective morphism \(\varphi \:A\to A_1(M)\) will be called a representation of \(A\) by continuous real functions. A representation is regular if it preserves the suprema (and the infima).
It is proved here that for any semisimple MV-algebra \(A\) there exists an extremal compact Hausdorff space \(M\) and a regular representation \(\varphi:A \to A_1(M)\).

06D35 MV-algebras
54C40 Algebraic properties of function spaces in general topology
54C30 Real-valued functions in general topology
54D30 Compactness
06F15 Ordered groups
Full Text: EuDML
[1] BIRKHOFF G.: Lattice Theory. (Rev.. Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc, New York, 1948. · Zbl 0033.10103
[2] CIGNOLI R.-D’OTTAVIANO I. M. L-MUNDICI D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic. Studia Logica Library Vol. 7, Kluwer Academic Publishers, Dordrecht, 2000. · Zbl 0937.06009
[3] CONRAD P.: Lattice Ordered Groups. Math. Res. Library, Tulane University, New Orleans, 1970. · Zbl 0258.06011
[4] DI NOLA A.-SESSA S.: On MV-algebras of continuous functions. Non-classical Logics and Their Application to Fuzzy Subsets (U. Hohle, E. P. Klement, Kluwer Academic Publishers, Dordrecht, 1996, pp. 22-31.
[5] GLUSCHANKOV D.: Cyclic ordered groups and MV-algebras. Czechoslovak Math. J. 43 (1993), 249-263.
[6] JAKUBfK J.: On complete MV-algebras. Czechoslovak Math. J. 45 (1995), 473-480. · Zbl 0841.06010
[7] JAKUBlK J.: On archimedean MV-algebras. Czechoslovak Math. J. 48 (1998), 575-582. · Zbl 0951.06011
[8] JAKUBIK J.: Complete distributivity of lattice ordered groups and of vector lattices. Czechoslovak Math. J. · Zbl 0998.06013
[9] MUNDICI D.: Interpretation of AFC* -algebras in Lukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15-63. · Zbl 0597.46059
[10] SIKORSKI R.: Boolean Algebras. (2nd, Springer-Verlag, Berlin-Goettingen-Heidelberg-New York, 1964. · Zbl 0123.01303
[11] VULIKH B. Z.: Introduction to the Theory of Semiordered Spaces. Gos. Izd. Fiz.-Mat. Lit., Moskva, 1961 [ · Zbl 0101.08501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.