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.

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

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.

Keywords: basic algebra; commutative basic algebra; symmetric basic algebra; state; homomorphism

Classification (MSC 2010): 03G25, 06D35, 06C15

DOI: 10.21136/MB.2016.0056-14

Full text available as PDF.


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


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