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Keywords:
basic algebra; commutative basic algebra; symmetric basic algebra; state; homomorphism
basic algebra; commutative basic algebra; symmetric basic algebra; state; homomorphism
Summary:
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:
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[2] Chajda, I.: 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] Chajda, I., Halaš, R.: On varieties of basic algebras. Soft Comput. 19 (2015), 261-267. DOI 10.1007/s00500-014-1365-y | Zbl 06654984
[4] Kalmbach, G.: Orthomodular Lattices. London Mathematical Society Monographs 18. Academic Press, London (1983). MR 0716496 | Zbl 0512.06011
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