Soft Computing

, Volume 23, Issue 24, pp 12929–12935 | Cite as

On the variety of Gödel MV-algebras

  • Antonio Di Nola
  • Revaz Grigolia
  • Gaetano VitaleEmail author


We introduce a new algebraic structure
$$\begin{aligned} (A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1) \end{aligned}$$
called GödelMV-algebra (GMV-algebra) such that
  • \((A, \otimes , \oplus , *, 0, 1)\) is MV-algebra;

  • \((A,\vee , \wedge ,\rightharpoonup , 0, 1)\) is a Gödel algebra (i. e. Heyting algebra satisfying the identity \((x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1\)).

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 .


MV-algebra Gödel algebra Many-valued logic 


Compliance with ethical standards

Conflict of interest

Authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of SalernoFiscianoItaly
  2. 2.I.I.A.S.S. “E. R. Caianiello”Vietri sul MareItaly
  3. 3.University of TbilisiTbilisiGeorgia

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