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basic algebra; monadic basic algebra; existential quantifier; universal quantifier; lattice with section antitone involution
The concept of monadic MV-algebra was recently introduced by A. Di Nola and R. Grigolia as an algebraic formalization of the many-valued predicate calculus described formerly by J. D. Rutledge [9]. This was also genaralized by J. Rachůnek and F. Švrček for commutative residuated $\ell $-monoids since MV-algebras form a particular case of this structure. Basic algebras serve as a tool for the investigations of much more wide class of non-classical logics (including MV-algebras, orthomodular lattices and their generalizations). This motivates us to introduce the monadic basic algebra as a common generalization of the mentioned structures.
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[2] Chajda I., Halaš R.: A basic algebra is an MV-algebra if and only if it is a BCC-algebra. Intern. J. Theor. Phys., to appear. MR 2377053 | Zbl 1145.06003
[3] 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
[4] Chajda I., Halaš R., Kühr J.: Many-valued quantum algebras. Algebra Universalis, to appear. MR 2480632 | Zbl 1219.06013
[5] Chajda I., Halaš R., Kühr J.: Semilattice Structures. : Heldermann Verlag, Lemgo, Germany. 2007. MR 2326262
[6] Chajda I., Kolařík M.: Independence of axiom system of basic algebras. Soft Computing, to appear, DOI 10.1007/s00500-008-0291-2. Zbl 1178.06007
[7] Di Nola A., Grigolia R.: On monadic MV-algebras. Ann. Pure Appl. Logic 128 (2006), 212–218. MR 2060551 | Zbl 1052.06010
[8] Rachůnek J., Švrček F.: Monadic bounded commutative residuated $\ell $-monoids. Order, to appear. MR 2425951 | Zbl 1151.06008
[9] Rutledge J. D.: On the definition of an infinitely-many-valued predicate calculus. J. Symbolic Logic 25 (1960), 212–216. MR 0138549 | Zbl 0105.00501
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