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Representable good EQ-algebras

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Abstract

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 Novák (Proceedings of the Czech-Japan seminar, ninth meeting, Kitakyushu and Nagasaki, 18–22 August, 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 lattice. One of the outcomes is the possibility to relax the commutativity of the multiplication. This has been elaborated by El-Zekey et al. (Fuzzy Sets Syst 2009, 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 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 run on lines of van Alten’s (J Algebra 247:672–691, 2002) characterization of representable integral residuated lattices. We also supply a number of potentially useful results, leading to this characterization.

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Notes

  1. Given an algebra \(\left\langle E,F\right\rangle ,\) where F is the set of operations on E, and \(F'\subseteq F.\) Then the algebra \(\left\langle E, F'\right\rangle \) is called the F′-reduct of \(\left\langle E,F\right\rangle .\) The subalgebras of \(\left\langle E,F'\right\rangle \) are then referred to as F′-subreducts of \(\left\langle E,F\right\rangle .\)

  2. This approach is well known in literature, see, e.g., Abdel-Hamid and Morsi (2007) where the authors established a representation theorem of prelinear residuated algebras, in which the lattice structure is not assumed.

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Acknowledgments

I would like to thank to Vilem Novák for the repeated reading of the drafts of this paper and helping me to improve it a lot in many aspects. Thanks are also due to the grant obtained from SAIA under the National Scholarship Program (NSP) of the Slovak Republic, and the Department of Mathematics and Descriptive Geometry at Slovak University of Technology at Bratislava, which provided excellent facilities for the duration of this research.

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  1. Department of Basic Sciences, High Institute of Technology, Benha University, Benha, 13512, Kalyobia, Egypt

    Moataz El-Zekey

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  1. Moataz El-Zekey
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Correspondence to Moataz El-Zekey.

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El-Zekey, M. Representable good EQ-algebras. Soft Comput 14, 1011–1023 (2010). https://doi.org/10.1007/s00500-009-0491-4

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