Abstract
Commutative basic algebras are non-associative generalizations of MV-algebras and form an algebraic semantics of a non-associative generalization of the propositional infinite-valued Łukasiewicz logic. In the paper we investigate additive closure and multiplicative interior operators on commutative basic algebras as a generalization of topological operators.
This work was supported by Palacký University IGA PrF 2014016 and IGA PrF 2015010 and by ESF Project CZ.1.07/2.3.00/20.0051.
Acknowledgement
The authors are very indebted to the anonymous referee for his/her interesting remarks and suggestions.
References
[1] Botur, M.—Halaš, R.: Commutative basic algebras and non-associative fuzzy logics, Arch. Math. Logic 48 (2009), 243–255.10.1007/s00153-009-0125-7Search in Google Scholar
[2] Botur, M.—Halaš, R.—Kühr, J.: States on commutative basic algebras, Fuzzy Sets and Systems 187 (2012), 77–91.10.1016/j.fss.2011.07.010Search in Google Scholar
[3] Botur, M.—Chajda, I.—Halaš, R.: Are basic algebras residuated structures?, Soft Comput 14 (2010), 251–255.10.1007/s00500-009-0399-zSearch in Google Scholar
[4] Chajda, I.—Halaš, R.—Kühr, J.: Many valued quantum algebras, Algebra Universalis 60 (2009), 63–90.10.1007/s00012-008-2086-9Search in Google Scholar
[5] Chajda, I.—Kühr, J.: Ideals and congruences of basic algebras, Soft Comput 17 (2013), 401–410.10.1007/s00500-012-0915-4Search in Google Scholar
[6] Galatos, N.—Jipsen, P.—Kowalski, T.—Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, Amsterdam, 2007.Search in Google Scholar
[7] Jipsen, P.—Tsinakis, C.: A survey of residuated lattices. In: Ordered Algebraic Structures (J. Martinez, ed.), Kluwer, Dordrecht, 2006, pp. 19–56.Search in Google Scholar
[8] Rachůnek, J.—Šalounová, D.: State operators on commutative basic algebras, WCCI 2012 IEEE World Congress on Computational Intelligence, June 10-15, 2012, Brisbane, Australia, 1511–1516.Search in Google Scholar
[9] Rachůnek, J.—Švrček, F.: MV-algebras with additive closure operators, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 39 (2000), 183–189.Search in Google Scholar
[10] Raswiowa, H.—Sikorski, R.: The Mathematics of Metamathematics, Panstw. Wyd. Nauk, Warszawa, 1963.Search in Google Scholar
© 2017 Mathematical Institute Slovak Academy of Sciences
