Abstract
In this paper we define the monadic pseudo BE-algebras and investigate their properties. We prove that the existential and universal quantifiers of a monadic pseudo BE-algebra form a residuated pair. Special properties are studied for the particular case of monadic bounded commutative pseudo BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is proved that the quantifiers on bounded commutative pseudo BE-algebras are also quantifiers on the corresponding pseudo MV-algebras. The monadic deductive systems and monadic congruences of monadic pseudo BE-algebras are defined and their properties are studied. It is proved that, in the case of a monadic distributive commutative pseudo BE-algebra there is a one-to-one correspondence between monadic congruences and monadic deductive systems, and the monadic quotient pseudo BE-algebra algebra is also defined.
(Communicated by Anatolij Dvurečenskij)
Acknowledgement
The author is very grateful to the anonimous referees for their useful remarks and suggestions on the subject that helped improving the presentation.
References
[1] Belluce, L. P.—Grigolia, R.—Lettieri, A.: Representations of monadic MV-algebras, Studia Logica 81 (2005), 123–144.10.1007/s11225-005-2805-6Search in Google Scholar
[2] Bezhanishvili, G.: Varieties of monadic Heyting algebras I, Studia Logica 61 (1998), 367–402.10.1023/A:1005073905902Search in Google Scholar
[3] Borzooei, R. A.—Borumand Saeid, A.—Rezaei, A.—Radfar, A.—Ameri, R.: On pseudo BE-algebras, Discuss. Math. Gen. Algebra Appl. 33 (2013), 95–108.10.7151/dmgaa.1193Search in Google Scholar
[4] Borzooei, R. A.—Borumand Saeid, A.—Rezaei, A.—Radfar, A.—Ameri, R.: Distributive pseudo BE-algebras, Fasc. Math. 54 (2015), 21–39.10.1515/fascmath-2015-0002Search in Google Scholar
[5] Castaño, D.—Cimadamore, C.—Díaz Varela, J. P.—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.2016.12.007Search in Google Scholar
[6] Chajda, I.—Kolaříc, M.: Monadic basic algebras, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 47 (2008), 27–36.Search in Google Scholar
[7] Cimadamore, C.—Díaz Varela, J. P.: Monadic Wajsberg hoops, Rev. Un. Mat. Argentina 57 (2016), 63–83.Search in Google Scholar
[8] Ciungu, L. C.: Local pseudo BCK-algebras with pseudo-product, Math. Slovaca 61 (2011), 127–154.10.2478/s12175-011-0001-xSearch in Google Scholar
[9] Ciungu, L. C.—Kühr, J.: New probabilistic model for pseudo BCK-algebras and pseudo-hoops, J. Mult.-Valued Logic Soft Comput. 20 (2013), 373–400.Search in Google Scholar
[10] Ciungu, L. C.: Non-commutative Multiple-Valued Logic Algebras, Springer, Cham, Heidelberg, New York, Dordrecht, London, 2014.10.1007/978-3-319-01589-7Search in Google Scholar
[11] Ciungu, L. C.: Commutative pseudo BE-algebras, Iran. J. Fuzzy Syst. 13(1) (2016), 131–144.Search in Google Scholar
[12] Di Nola, A.—Grigolia, R.: On monadic MV-algebras, Ann. Pure Appl. Logic 128 (2004), 125–139.10.1016/j.apal.2003.11.031Search in Google Scholar
[13] Galatos, N.—Jipsen, P.—Kowalski, T.—Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, New York, 2007.Search in Google Scholar
[14] Georgescu, G.—Iorgulescu, A.—Leuştean, I.: Monadic and closure MV-algebras, J. Mult.-Valued Logic Soft Comput. 3 (1998), 235–257.Search in Google Scholar
[15] Georgescu, G.—Iorgulescu, A.: Pseudo MV-algebras, J. Mult.-Valued Logic Soft Comput. 6 (2001), 95–135.Search in Google Scholar
[16] Georgescu, G.—Iorgulescu, A.: Pseudo-BCK algebras: An extension of BCK-algebras. Proceedings of DMTCS’01: Combinatorics, Computability and Logic, Springer, London, 2001, pp. 97–114.10.1007/978-1-4471-0717-0_9Search in Google Scholar
[17] Georgescu, G.—Leuştean, L.—Preoteasa, V.: Pseudo-hoops, J. Mult.-Valued Logic Soft Comput. 11 (2005), 153–184.Search in Google Scholar
[18] Ghorbani, S.: Monadic pseudo-equality algebras, Soft Comput. (2019), https://doi.org/10.1007/s00500-019-04243-5.10.1007/s00500-019-04243-5Search in Google Scholar
[19] Halmos, P.: Algebraic Logic, Chelsea Publ. Co, New York, 1962.Search in Google Scholar
[20] Imai, Y.—Iséki, K.: On axiom systems of propositional calculi. XIV, Proc. Japan Acad. 42 (1966), 19–22.10.3792/pja/1195522169Search in Google Scholar
[21] Iorgulescu, A.: Classes of pseudo-BCK algebras - Part I, J. Mult.-Valued Logic Soft Comput. 12 (2006), 71–130.Search in Google Scholar
[22] Iorgulescu, A.: Algebras of Logic as BCK-algebras, ASE Ed., Bucharest, 2008.Search in Google Scholar
[23] Iorgulescu, A.: Implicative-Groups vs. Groups and Generalizations, Matrix Rom Ed., Bucharest, 2018.Search in Google Scholar
[24] Iorgulescu, A.: Monadic involutive pseudo-BCK algebras, Acta Univ. Apulensis Math. Inform. 15 (2008), 159–178.Search in Google Scholar
[25] Kim, H. S.—Kim, Y. H.: On BE-algebras, Sci. Math. Jpn. 66 (2007), 113–116.Search in Google Scholar
[26] Kim, K. H.—Yon, Y. H.: Dual BCK-algebra and MV-algebra, Sci. Math. Jpn. 66 (2007), 247–254.Search in Google Scholar
[27] Kondo, M.: On residuated lattices with universal quantifiers, Bull. Iranian Math. Soc. 41 (2015), 923–929.Search in Google Scholar
[28] KüHR, J.: Pseudo-BCK semilattices, Demonstratio Math. 40 (2007), 495–516.10.1515/dema-2007-0302Search in Google Scholar
[29] KüHR, J.: Pseudo-BCK algebras and related structures, Univerzita Palackého v Olomouci, 2007.Search in Google Scholar
[30] KüHR, J.: Commutative pseudo BCK-algebras, Southeast Asian Bull. Math. 33 (2009), 451–475.Search in Google Scholar
[31] Rachůnek, J.: A non-commutative generalization of MV-algebras, Czechoslovak Math. J. 52 (2002), 255–273.10.1023/A:1021766309509Search in Google Scholar
[32] Rachůnek, J.—Šalounová, D.: Monadic bounded commutative residuated ℓ-monoids, Order 25 (2008), 157–175.10.1007/s11083-008-9088-2Search in Google Scholar
[33] Rachůnek, J.—Šalounová, D.: Monadic bounded residuated lattices, Order 30 (2013), 195–210.10.1007/s11083-011-9236-ySearch in Google Scholar
[34] Rachůnek, J.—Šalounová, D.: Monadic GMV-algebras, Arch. Math. Logic 47 (2008), 277–297.10.1007/s00153-008-0086-2Search in Google Scholar
[35] Rezaei, A.—Borumand Saeid, A.—Radfar, A.—Borzooei, R. A.: Congruence relations on pseudo BE-algebras, An. Univ. Craiova Math. Comp. Sci. Ser. 41 (2014), 166–176.10.29252/hatef.jahla.1.2.4Search in Google Scholar
[36] Rutledge, J. D.: A Preliminary Investigation of the Infinitely Many-valued Predicate Calculus, Ph.D. Thesis, Cornell University, 1959.Search in Google Scholar
[37] Walendziak, A.: On commutative BE-algebras, Sci. Math. Jpn. 69 (2009), 281–284.Search in Google Scholar
[38] Wang, J.—Xin, X.—He, P.: Monadic bounded hoops, Soft Comput. 22 (2018), 1749–1762.10.1007/s00500-017-2648-xSearch in Google Scholar
[39] Wang, J.—Xin, X.—He, P.: Monadic NM-algebras, Log. J. IGPL, https://doi.org/10.1093/jigpal/jzz005.10.1093/jigpal/jzz005Search in Google Scholar
[40] Xin, X.—Fu, Y.—Lai, Y.—Wang, J.: Monadic pseudo BCI-algebras and corresponding logics, Soft Comput. 23 (2019), 1499–1510.10.1007/s00500-018-3189-7Search in Google Scholar
[41] Zaheriani, S. Y.—Zahiri, O.: Monadic BE-algebras, J. Intell. Fuzzy Syst. 27 (2014), 2987–2995.10.3233/IFS-141257Search in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
