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Independence of axiom system of basic algebras. (English) Zbl 1178.06007

A basic algebra is an algebra \((A;\oplus,\neg, 0)\) of type \((2,1,0)\) satisfying the following axioms:
(BA1) \(x\oplus 0= x\),
(BA2) \(\neg\neg x= x\),
(BA3) \(x\oplus\neg 0= \neg 0= \neg 0\oplus x\),
(BA4) \(\neg(\neg x\oplus y)\oplus y= \neg(\neg y\oplus x)\oplus x\),
(BA5) \(\neg(\neg(\neg(x\oplus y)\oplus y)\oplus z)\oplus (x\oplus z)= \neg 0\).
This concept was first defined in the book [I. Chajda, R. Halaš and J. Kühr, Semilattice structures. Research and Exposition in Mathematics 30. Lemgo: Heldermann Verlag (2007; Zbl 1117.06001)] and generalizes the well-known concept of an MV-algebra (implicitly, such basic algebras were already studied in the paper [I. Chajda and P. Emanovský, Discuss. Math., Gen. Algebra Appl. 24, No. 1, 31–42 (2004; Zbl 1082.03055)]).
In the present paper, it is shown that (BA3) follows already from the other axioms, i.e., an algebra \((A;\oplus,\neg, 0)\) is a basic algebra if and only if it satisfies the axioms (BA1), (BA2), (BA4) and (BA5). Furthermore, it is shown that these remaining axioms are independent, and another system of axioms is given, not containing the axiom (BA2) of double negation.

MSC:

06D35 MV-algebras
03B52 Fuzzy logic; logic of vagueness
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References:

[1] Botur M, Halaš R (2008) Finite commutative basic algebras are MV-algebras. Multiple-valued Logic Soft Comput 14(1–2): 69–80 · Zbl 1236.06007
[2] Cattaneo G, Lombardo F (1998) Independent axiomatization of MV-algebras. Tatra Mountains Math Publ 15(2): 227–232 · Zbl 0939.03076
[3] Chajda I, Emanovský P (2004) Bounded lattices with antitone involutions and properties of MV-algebras. Discuss Math Gen Algebra Appl 24: 31–42 · Zbl 1082.03055
[4] Chajda I, Halaš R (2008) A basic algebra is an MV-algebra if and only if it is a BCC-algebra. Intern J Theor Phys 47(1): 261–267 · Zbl 1145.06003
[5] Chajda I, Halaš R, Kühr J (2007) Semilattice structures. Heldermann Verlag, Lemgo
[6] Chajda I, Halaš R, Kühr J (2008) Many-valued quantum algebras. Algebra Universalis (in press)
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