## Functorial representation theorems for MV$$_\Delta$$ algebras with additional operators.(English)Zbl 0987.06012

Building on his paper “An algebraic approach to propositional fuzzy logic” [J. Logic Lang. Inf. 9, No. 1, 91-124 (2000; Zbl 0942.06006)], in this dense paper the author establishes functorial representations for various varieties of MV-algebras enriched with extra operations. The prototypical representation of this kind was given by the present reviewer in J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059), by showing that MV-algebras are categorically equivalent to Abelian lattice-ordered groups with a strong unit. As is well known, the variety of MV-algebras is generated by the unit real interval $$[0,1]$$ equipped with negation $$1-x$$ and truncated sum. As a further natural operation over $$[0,1]$$ one may consider, e.g., the characteristic function $$\Delta$$ of the singleton 1. The resulting variety is denoted $$\text{MV}_\Delta$$. By adding the product operation (resp., product and its residuum, which amounts to truncated division) one further obtains $$\text{PMV}_\Delta$$ algebras (resp., $$L\Pi$$ algebras). The author establishes various categorical equivalences between these varieties and other mathematical structures. For instance, it is proved that the variety of $$L\Pi$$ algebras is categorically equivalent to regular commutative $$f$$-rings with unit and a suitable ideal. For background on MV-algebras see the monograph: R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning [Trends in Logic, Studia Logica Library, Vol. 7, Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)].

### MSC:

 06D35 MV-algebras 03E72 Theory of fuzzy sets, etc. 03B52 Fuzzy logic; logic of vagueness

### Citations:

Zbl 0942.06006; Zbl 0597.46059; Zbl 0937.06009
Full Text:

### References:

 [1] Alsina, C.; Trillas, E.; Valverde, L., On some logical connectives for fuzzy set theory, J math. anal. appl., 93, 15-26, (1983) · Zbl 0522.03012 [2] Baaz, M., Infinite-valued Gödel logics with 0-1 projections and relativizations, (), 23-33 · Zbl 0862.03015 [3] Balbes, R.; Dwinger, Ph., Distributive lattices, (1974), Univ. Missouri Press Columbia · Zbl 0321.06012 [4] Bigard, A.; Keimel, K.; Wolfenstein, S., Groupes et anneaux reticulès, Lecture notes in mathematics, 608, (1977), Springer-Verlag Berlin [5] Birkhoff, G.; Pierce, R.S., Lattice ordered rings, Anais acad. brasil. ci., 28, 41-69, (1956) · Zbl 0070.26602 [6] Burris, S.; Sankappanavar, H.P., A course in universal algebra, (1981), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0478.08001 [7] Chang, C.C., Algebraic analysis of many-valued logics, Trans. amer. math. soc., 88, 467-490, (1958) · Zbl 0084.00704 [8] Chang, C.C., A new proof of the completeness of łukasiewicz axioms, Trans amer. math. soc., 93, 74-80, (1959) · Zbl 0093.01104 [9] Cignoli, R.; Mundici, D., An invitation to Chang’s MV algebras, (), 171-197 · Zbl 0935.06010 [10] Cignoli, R.; Mundici, D., An elementary proof of Chang’s completeness theorem for the infinite-valued calculus of łukasiewicz, Studia logica, 58, 79-97, (1997) · Zbl 0876.03011 [11] Cignoli, R.; D’Ottaviano, I.M.L.; Mundici, D., Algebras das logicas de łukasiewicz, (1995) [12] Esteva, F.; Godo, L., Putting łukasiewicz and product logics together, Proceedings of ESTYLF’98, (September 1998), p. 339-346 [13] F. Esteva, L. Godo, and, F. Montagna, The ŁΠ and ŁΠ$$12$$ logics: two complete fuzzy systems joining Łukasiewicz and product logics, Arch. Math. Logic, to appear. · Zbl 0966.03022 [14] Gödel, K., Zum intuitionistischen aussagenkalkül, Anzeiger akademie der wissenschaften wien, math-naturwissensch, 69, (1932), p. 65-66 [15] Goodearl, K.R., Partially ordered abelian groups with interpolation, Mathematical surveys and monographs, 20, (1986), Am. Math. Soc Providence [16] Gumm, H.P.; Ursini, A., Ideals in universal algebra, Algebra universalis, 19, 45-54, (1984) · Zbl 0547.08001 [17] Hajek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030 [18] Keimel, K., Some trends in lattice ordered groups and rings, (), 131-161 · Zbl 0838.06015 [19] McKenzie, R., On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model, J. symbolic logic, 40, 186-196, (1975) · Zbl 0316.02052 [20] Mackenzie, R.; McNulty, G.; Taylor, W., Algebras, lattices, varieties, (1987), Wadsworth and Brooks/Cole Monterey [21] Montagna, F., An algebraic approach to propositional fuzzy logic, J. logic lang. inform., 9, 91-124, (2000) · Zbl 0942.06006 [22] Mundici, D., Interpretation of AFC* algebras in łukasiewicz sentential calculus, J. funct. anal., 65, 15-63, (1986) · Zbl 0597.46059 [23] Panti, G., A geometric proof of the completeness of the calculus of łukasiewicz, J. symbolic logic, 60, 563-578, (1995) · Zbl 0837.03018 [24] Pavelka, J., On fuzzy logic, I, II, III, Z. math. log. grundlagen math., 25, 45-52, (1979) · Zbl 0435.03020 [25] Takeuti, G.; Titani, S., Fuzzy logic and fuzzy set theory, Arch. math. logic, 32, 1-32, (1992) · Zbl 0786.03039 [26] Torrens, A., Boolean products of CW algebras and pseudocomplementation, Rep. math. logic, 23, 31-38, (1989) · Zbl 0741.06009
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