An algebraic study of the logic S5’(BL)
-
Juntao Wang
,
Xiaoli He
Abstract
P. Hájek introduced an S5-like modal fuzzy logic S5(BL) and showed that is equivalent to the monadic basic predicate logic mBL∀. Inspired by the above important results, D. Castaño et al. introduced monadic BL-algebras and their corresponding propositional logic S5’(BL), which is a simplified set of axioms of S5(BL). In this paper, we review the algebraic semantics of S5’(BL) and obtain some new results regarding to monadic BL-algebras. First we recall that S5’(BL) is completeness with respect to the variety 𝕄𝔹𝕃 of monadic BL-algebras and obtain a necessary and sufficient condition for the logic S5’(BL) to be semilinear. Then we study some further algebraic properties of monadic BL-algebras and discuss the relationship between monadic MV-algebras and monadic BL-algebras. Finally we give some characterizations of representable, simple, semisimple and directly indecomposable monadic BL-algebras, which are important members of the variety 𝕄𝔹𝕃. These results also constitute a crucial first step for providing an equivalent algebraic foundation for mBL∀.
This study was funded by a grant of National Natural Science Foundation of China (12001423,12001422,61976244, 11961016), the Natural Science Basic Research Plan in Shaanxi Province of China (2020JQ-762, 2021JQ-579) and the Natural Science Foundation of Education Committee of Shannxi Province (20JK0626).
(Communicated by Roberto Giuntini)
References
[1] Agliano, P.—Montagna, F.: Varieties of BL-algebras I: General properties, J. Pure Appl. Algebra 181 (2003), 105–129.10.1016/S0022-4049(02)00329-8Suche in Google Scholar
[2] Blok, W. J.—Pigozzi, D.: Algebraizable logics, Mem. Amer. Math. Soc. 396, vol 77, 1989.10.1090/memo/0396Suche in Google Scholar
[3] Borumd Saeid, A.—Somayeh, M.: Some results in BL-algebras, Math. Log. Q. 55 (2009), 649–658.10.1002/malq.200910025Suche in Google Scholar
[4] Buşneag, D.—Piciu, D.: On the lattice of deductive systems of a BL-algebra, Cent. Eur. J. Math. 2 (2003), 221–237.10.2478/BF02476010Suche in Google Scholar
[5] Burris, S.—Sankappanavar, M. P.: A Course in Universal Algebra, Springer-Verlag, New York, 1981.10.1007/978-1-4613-8130-3Suche in Google Scholar
[6] Castaño, D.—Cimadamore, C.— Verela, J. P. D.–Rueda, L.: Monadic BL-algebras: The equivalent algebraic semantics of Hájek’s monadic fuzzy logic, Fuzzy Sets and Systems 320 (2017), 40–59.10.1016/j.fss.2016.12.007Suche in Google Scholar
[7] Cintula, P.—Noguera, C.: Implicational (semilinear) logics III: completeness properties, Arch. Math. Logic 55 (2016), 353–372.10.1007/s00153-017-0577-0Suche in Google Scholar
[8] Di Nola, A.—Grigolia, R.: On monadic MV-algebras, Ann. Pure Appl. Logic 128 (2004), 125–139.10.1016/j.apal.2003.11.031Suche in Google Scholar
[9] Di Nola, A.—Grigolia, R.—Lenzi, G.: Topological spaces of monadic MV-algebras, Soft Comput. 23 (2019), 375–381.10.1007/s00500-018-3166-1Suche in Google Scholar
[10] Ghorbani, S: Monadic pseudo equality algebras, Soft Comput. 23 (2019), 12937–12950.10.1007/s00500-019-04243-5Suche in Google Scholar
[11] Grigolia, R.: Monadic BL-algebras, Georgian Math. J. 13 (2006), 267–276.10.1515/GMJ.2006.267Suche in Google Scholar
[12] Hájek, P.: Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.10.1007/978-94-011-5300-3Suche in Google Scholar
[13] Hájek, P.: Fuzzy logics with Noncommutative conjuctions, J. Logic Comput. 13 (2003), 469–479.10.1093/logcom/13.4.469Suche in Google Scholar
[14] Hájek, P.—Haniková, Z.: Interpreting lattice-valued set theory in fuzzy set theory, Log. J. IGPL 21 (2013), 77–90.10.1093/jigpal/jzs023Suche in Google Scholar
[15] Halmos, R. P.: Algebraic logic, I. Monadic boolean algebras, Composition Mathematica 12 (1955), 217–249.Suche in Google Scholar
[16] Liu, H. L.: On categorical equivalences of equality algebras and monadic equality algebras, Log. J. IGPL 27 (2019), 267–280.10.1093/jigpal/jzy038Suche in Google Scholar
[17] Noguera, C: Algebraic Study of Axiomatic Extensions of Triangular Norm Based Fuzzy Logics, Ph.D. thesis, IIIA-CSIC, 2006.Suche in Google Scholar
[18] Orellano Figallo, A.: A topological duality for moandic MV-algebras, Soft Comput. 21 (2017), 7119–7123.10.1007/s00500-016-2255-2Suche in Google Scholar
[19] Orellano Figallo, A.—Pascual, I.—Ziliani, A.: Monadic distributive lattices, Log. J. IGPL 15 (2007), 535–551.10.1093/jigpal/jzm039Suche in Google Scholar
[20] Orellano Figallo, A.—Pascual, I.: Topological dualities for strong monadic distributive lattices and applications, Journal of Multiple-Valued Logic and Soft Computing 32 (2019), 499–540.Suche in Google Scholar
[21] Rachůnek, J.—Şalounová, D.: Monadic GMV-algebras, Arch. Math. Logic 47 (2008), 277–297.10.1007/s00153-008-0086-2Suche in Google Scholar
[22] Rachůnek, J.—Şvrček, F.: Monadic bounded commutative residuated ℓ-monoids, Order 25 (2008), 157–175.10.1007/s11083-008-9088-2Suche in Google Scholar
[23] Rasiowa, H: An Algebraic Approach to Non-Classical Logic, North Holland, Amsterdam, 1974.Suche in Google Scholar
[24] Rutledge, J. D.: A Preliminary Investigation of the Infinitely Many-Valued Predicate Calculus, Ph. D, Thesis, Cornell University, 1959.Suche in Google Scholar
[25] Turunen, E: BL-algebras of basic fuzzy logic, Mathware Soft Computing 6 (1998), 49–61.Suche in Google Scholar
[26] Wang, J. T.—He, P. F.—She, Y. H.: Monadic NM-algebras, Log. J. IGPL 27 (2019), 812–835.10.1093/jigpal/jzz005Suche in Google Scholar
[27] Wang, J. T.—Xin, X. L.—He, P. F.: Monadic bounded hoops, Soft Comput. 22 (2018), 1749–1762.10.1007/s00500-017-2648-xSuche in Google Scholar
[28] Wang, J. T.—Wang, M.: The lattices of monadic filters in monadic BL-algebras, IAENG Int. J. Appl. Math. Comput. Sci. 50 (2020), 656–660.Suche in Google Scholar
[29] Xin, X. L.—Fu, Y. L.—Lai, Y. Y.—Wang, J. T.: Monadic pseudo BCI-algebras and corresponding logics, Soft Comput. 23 (2019), 1499–1510.10.1007/s00500-018-3189-7Suche in Google Scholar
© 2022 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Prof. RNDr. Július korbaš, CSC. passed away
- Kalmbach measurability In d0-algebras
- Quantifiers on L-algebras
- Ideals of functions with compact support in the integer-valued case
- An algebraic study of the logic S5’(BL)
- Triangular numbers and generalized fibonacci polynomial
- A general matrix series inversion pair and associated polynomials
- Quantum ostrowski type inequalities for pre-invex functions
- Hermite-Hadamard type inequalities for interval-valued fractional integrals with respect to another function
- Approximating families for lattice outer measures on unsharp quantum logics
- On a generalized Lamé-Navier system in ℝ3
- Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions
- On some classical properties of normed spaces via generalized vector valued almost convergence
- Fan-Hemicontinuity for the gradient of the norm in Hilbert space
- Solvability of mixed problems for heat equations with two nonlocal conditions
- The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow
- A note on set-star-K-Menger spaces
- A bivariate extension of the Omega distribution for two-dimensional proportional data
- Bernstein polynomials based iterative method for solving fractional integral equations
- The symmetric 4-Player gambler’s problem with unequal initial stakes
- Enveloping action: Convergence spaces
Artikel in diesem Heft
- Prof. RNDr. Július korbaš, CSC. passed away
- Kalmbach measurability In d0-algebras
- Quantifiers on L-algebras
- Ideals of functions with compact support in the integer-valued case
- An algebraic study of the logic S5’(BL)
- Triangular numbers and generalized fibonacci polynomial
- A general matrix series inversion pair and associated polynomials
- Quantum ostrowski type inequalities for pre-invex functions
- Hermite-Hadamard type inequalities for interval-valued fractional integrals with respect to another function
- Approximating families for lattice outer measures on unsharp quantum logics
- On a generalized Lamé-Navier system in ℝ3
- Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions
- On some classical properties of normed spaces via generalized vector valued almost convergence
- Fan-Hemicontinuity for the gradient of the norm in Hilbert space
- Solvability of mixed problems for heat equations with two nonlocal conditions
- The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow
- A note on set-star-K-Menger spaces
- A bivariate extension of the Omega distribution for two-dimensional proportional data
- Bernstein polynomials based iterative method for solving fractional integral equations
- The symmetric 4-Player gambler’s problem with unequal initial stakes
- Enveloping action: Convergence spaces
