Skip to content
Accessible Unlicensed Requires Authentication Published by De Gruyter February 28, 2017

Interior and closure operators on commutative basic algebras

Jiří Rachůnek and Zdeněk Svoboda
From the journal Mathematica Slovaca

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.

MSC 2010: 03G05; 03G10; 06D35; 06A15

(Communicated by Anatolij Dvurečenskij)


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.Search in Google Scholar

[2] Botur, M.—Halaš, R.—Kühr, J.: States on commutative basic algebras, Fuzzy Sets and Systems 187 (2012), 77–91.Search in Google Scholar

[3] Botur, M.—Chajda, I.—Halaš, R.: Are basic algebras residuated structures?, Soft Comput 14 (2010), 251–255.Search in Google Scholar

[4] Chajda, I.—Halaš, R.—Kühr, J.: Many valued quantum algebras, Algebra Universalis 60 (2009), 63–90.Search in Google Scholar

[5] Chajda, I.—Kühr, J.: Ideals and congruences of basic algebras, Soft Comput 17 (2013), 401–410.Search 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

Received: 2014-7-2
Accepted: 2015-4-17
Published Online: 2017-2-28
Published in Print: 2017-2-1

© 2017 Mathematical Institute Slovak Academy of Sciences