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lattice with section antitone involution; basic algebra; commutative basic algebra; MV-algebra
A basic algebra is an algebra of the same type as an MV-algebra and it is in a one-to-one correspondence to a bounded lattice having antitone involutions on its principal filters. We present a simple criterion for checking whether a basic algebra is commutative or even an MV-algebra.
[1] Botur, M., Halaš, R.: Finite commutative basic algebras are MV-algebras. (to appear) in Multiple-Valued Logic and Soft Computing.
[2] Chajda, I.: Lattices and semilattices having an antitone involution in every upper interval. Comment. Math. Univ. Carol. 44 (2003), 577-585. MR 2062874 | Zbl 1101.06003
[3] Chajda, I., Emanovský, P.: Bounded lattices with antitone involutions and properties of MV-algebras. Discuss. Math., Gener. Algebra Appl. 24 (2004), 31-42. DOI 10.7151/dmgaa.1073 | MR 2117673 | Zbl 1082.03055
[4] Chajda, I., Halaš, R.: A basic algebra is an MV-algebra if and only if it is a BCC-algebra. Int. J. Theor. Phys. 47 (2008), 261-267. DOI 10.1007/s10773-007-9468-1 | MR 2377053 | Zbl 1145.06003
[5] Chajda, I., Halaš, R., Kühr, J.: Distributive lattices with sectionally antitone involutions. Acta Sci. Math. (Szeged) 71 (2005), 19-33. MR 2160352 | Zbl 1099.06006
[6] Cignoli, R. L. O., D'Ottaviano, M. L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Acad. Publ., Dordrecht (2000). MR 1786097 | Zbl 0937.06009
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