Representable good EQ-algebras. (English) Zbl 1201.03061

Summary: Recently, a special algebra, called EQ-algebra (we call it here commutative EQ-algebra since its multiplication is assumed to be commutative), has been introduced by V. Novák [“EQ-algebras: primary concepts and properties”, in: Proceedings of the Czech-Japan seminar, ninth meeting, Kitakyushu and Nagasaki, 18–22 August 2006, Graduate School of Information, Waseda Univ., 219–223 (2006)], which aims at becoming the algebra of truth values for fuzzy-type theory. Its implication and multiplication are no more closely tied by the adjunction and so this algebra generalizes commutative residuated lattices. One of the outcomes is the possibility to relax the commutativity of the multiplication. This has been elaborated by the author, V. Novák and R. Mesiar [“Semicopula-based EQ-algebras”, Fuzzy Sets Syst. (submitted)]. We continue in this paper the study of EQ-algebras (i.e., those with multiplication not necessarily commutative). We introduce prelinear EQ-algebras in which the join-semilattice structure is not assumed. We show that every prelinear and good EQ-algebra is a lattice EQ-algebra. Moreover, the \(\{\wedge,\vee,\rightarrow,1\}\)-reduct of a prelinear and separated lattice EQ-algebra inherits several lattice-related properties from the product of linearly ordered and separated EQ-algebras. We show that prelinearity alone does not characterize the representable class of all good (commutative) EQ-algebras. One of the main results of this paper is to characterize the representable good EQ-algebras. This is mainly based on the fact that \(\{\rightarrow,1\}\)-reducts of good EQ-algebras are BCK-algebras and is performed similarly to C. J. van Alten’s [J. Algebra 247, No. 2, 672–691 (2002; Zbl 1001.06012)] characterization of representable integral residuated lattices. We also supply a number of potentially useful results, leading to this characterization.


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


Zbl 1001.06012
Full Text: DOI


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