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Erschienen in:
22.07.2019 | Foundations
On the variety of Gödel MV-algebras
Erschienen in: Soft Computing | Ausgabe 24/2019
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Abstract
We introduce a new algebraic structure called Gödel–MV-algebra (GMV-algebra) such that It is shown that the lattice of congruences of a GMV -algebra \((A, \otimes , \oplus , *, \rightharpoonup , 0, 1)\) is isomorphic to the lattice of Skolem filters (i. e. special type of MV-filters) of the MV-algebra \((A, \otimes , \oplus , *, 0, 1)\). Any GMV-algebra is bi-Heyting algebra. Any chain GMV-algebra is simple, and any GMV-algebra is semi-simple. Finitely generated GMV-algebras are described, and finitely generated finitely presented GMV-algebras are characterized. The algebraic counterpart of axiomatically presented GMV-logic is GMV-algebras
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$$\begin{aligned} (A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1) \end{aligned}$$
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\((A, \otimes , \oplus , *, 0, 1)\) is MV-algebra;
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\((A,\vee , \wedge ,\rightharpoonup , 0, 1)\) is a Gödel algebra (i. e. Heyting algebra satisfying the identity \((x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1\)).