On good EQ-algebras. (English) Zbl 1242.03089

El-Zekey, Moataz; Novák, Vilém; Mesiar, Radko

Fuzzy Sets Syst. 178, No. 1, 1-23 (2011).

In this paper the authors investigate new properties of EQ-algebras and their special case of good EQ-algebras.
EQ-algebras have been introduced by Vilém Novák and they have three binary operations, meet, multiplication and fuzzy equality, and a unit element. The motivation of introducing these algebras is the development of fuzzy logic with the basic connective being a fuzzy equality instead of an implication. The notions of prefilter and filters are introduced and studied and good EQ-algebras are defined. In particular, it is shown that \(\{\rightarrow, 1\}\)-reducts of good EQ-algebras are BCK-algebras. The good EQ-algebras are enriched with a unary operation \(\Delta\), called Baaz delta, fulfilling some additional assumptions. A characterization theorem for the representable good EQ-algebras is also proved for the enriched algebra.
The main results of the paper are the following:

1.

The class of EQ-algebras is a variety.

2.

The \(\{\rightarrow, 1\}\)-reducts of good EQ-algebras are BCK-meet-semilattices.

3.

If \({\mathcal E}=(E, \wedge, \otimes, \sim, 1)\) is a residuated EQ-algebra, then its multiplication \(\otimes\) is commutative and \({\mathcal E}'=(E, \wedge, \otimes, \rightarrow, 1)\) is a commutative residuated lattice, where \(a\rightarrow b=(a\wedge b)\sim a\).

4.

If \({\mathcal E}_{\Delta}\) is an EQ-algebra, then the following are equivalent: (a) \({\mathcal E}_{\Delta}\) is representable; (b) \({\mathcal E}_{\Delta}\) satisfies \((a\rightarrow b)\rightarrow u\leq [(d\rightarrow (d\otimes (c\rightarrow ((b\rightarrow a)\otimes c))))\rightarrow u] \rightarrow u\); (c) \({\mathcal E}_{\Delta}\) satisfies \((d\rightarrow (d\otimes (c\rightarrow ((b\rightarrow a)\otimes c))))\rightarrow u \leq ((a\rightarrow b)\rightarrow u)\rightarrow u\); (d) \({\mathcal E}_{\Delta}\) is prelinear and every minimal prime prefilter of \({\mathcal E}_{\Delta}\) is a filter of \({\mathcal E}_{\Delta}\).

We conclude that the paper under review contains very interesting results and can be a starting point for future studies.

Reviewer: Lavinia Ciungu (Craiova)

MSC:

03G25	Other algebras related to logic
03B52	Fuzzy logic; logic of vagueness
06F35	BCK-algebras, BCI-algebras

Keywords:

EQ-algebras; fuzzy equality; fuzzy logic; residuated lattice; BCK-algebra; representable algebra

Software:

ETPS

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