Fuzzy deductive systems of RM algebras
Abstract
The theory of fuzzy deductive systems in RM algebras is developed. Various characterizations of fuzzy deductive systems are given. It is proved that the set of all fuzzy deductive systems of a RM algebra đť’ś is a complete lattice (it is distributive if đť’ś is a pre-BBBCC algebra). Some characterizations of Noetherian RM algebras by fuzzy deductive systems are obtained. In pre-BBBZ algebras, the fuzzy deductive system generated by a fuzzy set is constructed. Finally, closed fuzzy deductive systems are defined and studied. It is showed that in finite CI and pre-BBBZ algebras, every fuzzy deductive system is closed. Moreover, the homomorphic properties of (closed) fuzzy deductive systems are provided.
References
[1] Busneag, D.: A note on deductive systems of a Hilbert algebra, Kobe J. Math. 2 (1985), 29–35.Suche in Google Scholar
[2] Busneag, D.: Hilbert algebras of fractions and maximal Hilbert algebras of quotients, Kobe J. Math. 5 (1988), 161–172.Suche in Google Scholar
[3] Busneag, D.: Hertz algebras of fractions and maximal Hertz algebras of quotients, Math. Japonica 39 (1993), 461–469.Suche in Google Scholar
[4] Diego, A.: Sur les Algèbres de Hilbert. Collection de Logigue Math. Serie A, XXI, Gauthier-Villars, 1966.Suche in Google Scholar
[5] Dudek, W. A.—Karamdin, B.—Bhatti, S. A.: Branches and ideals of weak BCC-algebras, Algebra Colloq. 18 (Spec 1) (2011), 899–914.10.1142/S1005386711000782Suche in Google Scholar
[6] Dudek, W. A.—Zhang, X.—Wang, Y.: Ideals and atoms of BZ algebras, Math. Slovaca 59 (2009),387–404.10.2478/s12175-009-0135-2Suche in Google Scholar
[7] Henkin, L.: An algebraic characterization of quantifiers, Fund. Math. 37 (1950), 63–74.10.4064/fm-37-1-63-74Suche in Google Scholar
[8] Hoo, C. S.: Filters and ideals in BCI-algebras, Math. Japonica 36 (1991), 987–997.Suche in Google Scholar
[9] Hu, Q. P.—Li, X.: On BCH-algebras, Math. Semin. Notes 11 (1983), 313–320.Suche in Google Scholar
[10] Imai, Y.—Iséki, K.: On axiom systems of propositional calculi XIV, Proc. Jpn. Acad. 42 (1966), 19–22.10.3792/pja/1195522169Suche in Google Scholar
[11] Iorgulescu, A.: New generalizations of BCI, BCK and Hilbert algebras – Part I, J. Mult. Valued Logic Soft Comput. 27 (2016), 353–406.Suche in Google Scholar
[12] Iorgulescu, A.: Implicative-groups vs. Groups and Generalizations, Matrix Rom, Bucuresti, 2018.Suche in Google Scholar
[13] Iséki, K.: An algebra related with a propositional calculus, Proc. Jpn. Acad. 42 (1966), 26–29.10.3792/pja/1195522171Suche in Google Scholar
[14] Iséki, K.—Tanaka, S.: Ideal theory of BCK-algebras, Math. Japonica 21 (1976), 351–366.Suche in Google Scholar
[15] Jun, Y. B.—Hong, S. M.: Fuzzy deductive systems of Hilbert algebras, Indian J. Pure Appl. Math. 27 (1996), 141–151.Suche in Google Scholar
[16] Jun, Y. B.—Roh, E. H.—Kim, H. S.: On BH-algebras, Sci. Math. Jpn. 1 (1998), 347–354.Suche in Google Scholar
[17] Jun, Y. B.: Some results on ideals of BCK-algebras, Sci. Math. Jpn. e-2001, 411–414.Suche in Google Scholar
[18] Kim, K. H.: A note on CI-algebras, Int. Math. Forum 6 (2011), 1–5.Suche in Google Scholar
[19] Meng, B. L.: CI-algebra, Sci. Math. Jpn. 71 (2010), 11–17; e-2009, 695–701.Suche in Google Scholar
[20] Meng, B. L.: Closed filters in CI-algebras, Sci. Math. Jpn. e-2010, 265–270.Suche in Google Scholar
[21] Meng, J.: On ideals in BCK-algebras, Math. Japonica 40 (1994), 143–154.Suche in Google Scholar
[22] Meng, J.—Wei, S. M.: Periodic BCI-algebras and closed ideals, Math. Japonica 38 (1993), 571–575.Suche in Google Scholar
[23] Piekart, B.—Walendziak, A.: On filters and upper sets in CI-algebras, Algebra Discrete Math. 7 (2011), 97–103.Suche in Google Scholar
[24] Swamy, U. M.—Raju, D. V.: Algebraic fuzzy systems, Fuzzy Sets and Systems 41 (1991), 187–194.10.1016/0165-0114(91)90222-CSuche in Google Scholar
[25] Walendziak, A.: Deductive systems and congruences in RM algebras, J. Mult.-Valued Logic Soft Comput. 30 (2018), 521–539.Suche in Google Scholar
[26] Walendziak, A.: The property of commutativity for some generalizations of BCK algebras, Soft Comput. (2018), https://doi.org/10.1007/s00500-018-03691-910.1007/s00500-018-03691-9Suche in Google Scholar
[27] Walendziak, A.: Irreducible and prime deductive systems of some generalizations of BCK algebras, J. Mult.-Valued Logic Soft Comput. 34 (2020), 129–148.Suche in Google Scholar
[28] Ye, R.: On BZ-algebras. In: Selected papers on BCI/BCK-algebras and Computer Logic, Shanghai Jiaotong Univ., 1991, pp. 21–24.Suche in Google Scholar
[29] Zadeh, L. A.: Fuzzy sets, Inform. & Control 8 (1965), 338–353.10.21236/AD0608981Suche in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Fuzzy deductive systems of RM algebras
- Congruence pairs of principal MS-algebras and perfect extensions
- The lattices of 𝔏-fuzzy state filters in state residuated lattices
- Central lifting property for orthomodular lattices
- EQ-Modules
- On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
- Remarks on some generalization of the notion of microscopic sets
- Disjointness of composition operators on Hv0 spaces
- A common fixed point theorem for non-self mappings in strictly convex menger PM-spaces
- The Poincaré-Cartan forms of one-dimensional variational integrals
- Coarse cohomology with twisted coefficients
- Divisible extension of probability
- Asymptotic behavior of the records of multivariate random sequences in a norm sense
- Strong convergence of the functional nonparametric relative error regression estimator under right censoring
- A new kumaraswamy generalized family of distributions: Properties and applications
- Efficient message transmission via twisted Edwards curves
- Computation of several Hessenberg determinants
Artikel in diesem Heft
- Regular papers
- Fuzzy deductive systems of RM algebras
- Congruence pairs of principal MS-algebras and perfect extensions
- The lattices of 𝔏-fuzzy state filters in state residuated lattices
- Central lifting property for orthomodular lattices
- EQ-Modules
- On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
- Remarks on some generalization of the notion of microscopic sets
- Disjointness of composition operators on Hv0 spaces
- A common fixed point theorem for non-self mappings in strictly convex menger PM-spaces
- The Poincaré-Cartan forms of one-dimensional variational integrals
- Coarse cohomology with twisted coefficients
- Divisible extension of probability
- Asymptotic behavior of the records of multivariate random sequences in a norm sense
- Strong convergence of the functional nonparametric relative error regression estimator under right censoring
- A new kumaraswamy generalized family of distributions: Properties and applications
- Efficient message transmission via twisted Edwards curves
- Computation of several Hessenberg determinants