## On good EQ-algebras.(English)Zbl 1242.03089

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.

### MSC:

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

ETPS
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