##
**EQ-algebras.**
*(English)*
Zbl 1184.03067

In this paper, the authors introduce the concept of an EQ-algebra, which should become the algebra of truth values for a higher-order fuzzy logic (a fuzzy type theory, FTT). The motivation stems from the fact that, until now, the truth values in FTT were assumed to form either an MTL-, BL-, or MV-algebra, all of them being special kinds of residuated lattices in which the basic operations are the monoidal operation (multiplication) and its residuum.

The main operation of an EQ-algebra is a fuzzy equality \(\sim\) (a natural interpretation of the main connective in fuzzy type theory) accompanied by the binary operations of meet and multiplication (\(\otimes\)). The essential difference between residuated lattices and EQ-algebras lies in the definition of implication \(\rightarrow\). Unlike residuated lattices, where \(\rightarrow\) is joined with \(\otimes\) through the adjointness property, in EQ-algebras \(\rightarrow\) is defined directly from the fuzzy equality \(\sim\). Hence, the adjointness property, which strictly couples \(\rightarrow\) and \(\otimes\), is relaxed. This has as consequence that the multiplication can be non-commutative without forcing two kinds of implication. In this paper, the authors study basic properties of commutative EQ-algebras (i.e. the multiplication \(\otimes\) is commutative) and present several special kinds of EQ-algebras.

The relation between EQ-algebras and residuated lattices is quite intricate and it seems that the former open the door to another look at the latter. When considering implication only, it can be shown that the corresponding reducts of EQ-algebras are BCK-algebras, and so, residuated lattices are “hidden” inside. On the other hand, EQ-algebras form a variety and they are not equivalent with residuated lattices; in fact, EQ-algebras generalize residuated lattices because they relax the tie between multiplication and residuation (i.e. between conjunction and implication in logic).

The EQ-algebras open the door also to introducing a class of EQ-logics, which might cast a different light on fuzzy logics studied until now only from the point of view of the properties of implication.

The main operation of an EQ-algebra is a fuzzy equality \(\sim\) (a natural interpretation of the main connective in fuzzy type theory) accompanied by the binary operations of meet and multiplication (\(\otimes\)). The essential difference between residuated lattices and EQ-algebras lies in the definition of implication \(\rightarrow\). Unlike residuated lattices, where \(\rightarrow\) is joined with \(\otimes\) through the adjointness property, in EQ-algebras \(\rightarrow\) is defined directly from the fuzzy equality \(\sim\). Hence, the adjointness property, which strictly couples \(\rightarrow\) and \(\otimes\), is relaxed. This has as consequence that the multiplication can be non-commutative without forcing two kinds of implication. In this paper, the authors study basic properties of commutative EQ-algebras (i.e. the multiplication \(\otimes\) is commutative) and present several special kinds of EQ-algebras.

The relation between EQ-algebras and residuated lattices is quite intricate and it seems that the former open the door to another look at the latter. When considering implication only, it can be shown that the corresponding reducts of EQ-algebras are BCK-algebras, and so, residuated lattices are “hidden” inside. On the other hand, EQ-algebras form a variety and they are not equivalent with residuated lattices; in fact, EQ-algebras generalize residuated lattices because they relax the tie between multiplication and residuation (i.e. between conjunction and implication in logic).

The EQ-algebras open the door also to introducing a class of EQ-logics, which might cast a different light on fuzzy logics studied until now only from the point of view of the properties of implication.

Reviewer: Florentina Chirteş (Craiova)

### Keywords:

EQ-algebra; fuzzy equality; fuzzy logic; fuzzy type theory; higher-order fuzzy logic; residuated lattice### Software:

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\textit{V. Novák} and \textit{B. De Baets}, Fuzzy Sets Syst. 160, No. 20, 2956--2978 (2009; Zbl 1184.03067)

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### References:

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