Mathematica Bohemica, Vol. 142, No. 2, pp. 197-210, 2017
States on basic algebras
Ivan Chajda, Helmut Länger
Received August 11, 2014. First published December 12, 2016.
Abstract: States on commutative basic algebras were considered in the literature as generalizations of states on MV-algebras. It was a natural question if states exist also on basic algebras which are not commutative. We answer this question in the positive and give several examples of such basic algebras and their states. We prove elementary properties of states on basic algebras. Moreover, we introduce the concept of a state-morphism and characterize it among states. For basic algebras which are the certain pastings of Boolean algebras the construction of a state-morphism is shown.
References: [1] M. Botur, R. Halaš, J. Kühr: States on commutative basic algebras. Fuzzy Sets Syst. 187 (2012), 77-91. DOI 10.1016/j.fss.2011.07.010 | MR 2851997 | Zbl 1266.03070 [2] I. Chajda: Basic algebras and their applications. An overview. Proc. 81st Workshop on General Algebra (J. Czermak et al., eds.). Salzburg, Austria, 2011, Johannes Heyn, Klagenfurt (2012), 1-10. MR 2908429 | Zbl 1280.06004 [3] I. Chajda, R. Halaš: On varieties of basic algebras. Soft Comput. 19 (2015), 261-267. DOI 10.1007/s00500-014-1365-y | Zbl 06654984 [4] G. Kalmbach: Orthomodular Lattices. London Mathematical Society Monographs 18. Academic Press, London (1983). MR 0716496 | Zbl 0512.06011
Affiliations: Ivan Chajda, Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: ivan.chajda@upol.cz; Helmut Länger, TU Wien, Fakultät für Mathematik und Geoinformation, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria, e-mail: helmut.laenger@tuwien.ac.at
