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THE LOOMIS–SIKORSKI THEOREM FOR $EMV$ -ALGEBRAS

  • ANATOLIJ DVUREČENSKIJ (a1) (a2) and OMID ZAHIRI (a3)

Abstract

An EMV-algebra resembles an MV-algebra in which a top element is not guaranteed. For $\unicode[STIX]{x1D70E}$ -complete $EMV$ -algebras, we prove an analogue of the Loomis–Sikorski theorem showing that every $\unicode[STIX]{x1D70E}$ -complete $EMV$ -algebra is a $\unicode[STIX]{x1D70E}$ -homomorphic image of an $EMV$ -tribe of fuzzy sets where all algebraic operations are defined by points. To prove it, some topological properties of the state-morphism space and the space of maximal ideals are established.

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The first author is grateful for support from grants APVV-16-0073, VEGA no. 2/0069/16 SAV and GAČR 15-15286S.

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[1] Barbieri, G. and Weber, H., ‘Measures on clans and on MV-algebras’, in: Handbook of Measure Theory, Vol. II (ed. Pap, E.) (Elsevier Science, Amsterdam, 2002), 911945.
[2] Belluce, L. P., ‘Semisimple algebras of infinite valued logic and bold fuzzy set theory’, Can. J. Math. 38(6) (1986), 13561379.
[3] Chang, C. C., ‘Algebraic analysis of many valued logics’, Trans. Amer. Math. Soc. 88(2) (1958), 467490.
[4] Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of Many-Valued Reasoning (Springer Science and Business Media, Dordrecht, 2000).
[5] Conrad, P. and Darnel, M. R., ‘Generalized Boolean algebras in lattice-ordered groups’, Order 14(4) (1997), 295319.
[6] Di Nola, A. and Russo, C., ‘The semiring-theoretic approach to MV-algebras’, Fuzzy Sets and Systems 281 (2015), 134154.
[7] Dvurečenskij, A., ‘Loomis–Sikorski theorem for 𝜎-complete MV-algebras and -groups’, J. Aust. Math. Soc. 68(2) (2000), 261277.
[8] Dvurečenskij, A., ‘Pseudo MV-algebras are intervals in -groups’, J. Aust. Math. Soc. 72(3) (2002), 427446.
[9] Dvurečenskij, A. and Pulmannová, S., New Trends in Quantum Structures (Kluwer Academic Publishers, Dordrecht, Ister Science, Bratislava, 2000).
[10] Dvurečenskij, A. and Zahiri, O., ‘On EMV-algebras’, Preprint, 2017, arXiv:1706.00571.
[11] Galatos, N. and Tsinakis, C., ‘Generalized MV-algebras’, J. Algebra 283 (2005), 254291.
[12] Georgescu, G. and Iorgulescu, A., ‘Pseudo MV-algebras’, Mult.-Valued Logic 6(1–2) (2001), 95135.
[13] Goodearl, K. R., Partially Ordered Abelian Groups with Interpolation, Matematical Surveys and Monographs, 20 (American Mathematical Society, Providence, RI, 1986).
[14] Kelley, J. L., General Topology, Van Nostrand, Toronto 1957, Graduate Texts in Mathematics (reprinted by Springer, New York, 1975).
[15] Loomis, L. H., ‘On the representation of 𝜎-complete Boolean algebras’, Bull. Amer. Math. Soc. (N.S.) 53(8) (1947), 757760.
[16] Luxemburg, W. A. J. and Zaanen, A. C., Riesz Spaces, Vol. 1 (North-Holland Publishers, Amsterdam, London, 1971).
[17] Mundici, D., ‘Interpretation of AF C -algebras in Łukasiewicz sentential calculus’, J. Funct. Anal. 65(1) (1986), 1563.
[18] Mundici, D., ‘Tensor products and the Loomis–Sikorski theorem for MV-algebras’, Adv. Appl. Math. 22(2) (1999), 227248.
[19] Sikorski, R., ‘On the representation of Boolean algebras as fields of sets’, Fundam. Math. 35(1) (1948), 247258.
[20] Stone, M. H., ‘Applications of the theory of Boolean rings to general topology’, Trans. Amer. Math. Soc. 41(3) (1937), 375481.
[21] Stone, M. H., ‘Topological representations of distributive lattices and Brouwerian logics’, Časopis pro pěstování matematiky a fysiky 67(1) (1938), 125.
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