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Preideals in EQ-algebras

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Abstract

EQ-algebras were introduced by Novak (2006) as an algebraic structure of truth values for fuzzy-type theory (FFT). Novák and De Baets (2009) introduced various kinds of EQ-algebras such as good, residuated, and IEQ-algebras. In this paper, we define the notion of (pre)ideal in bounded EQ-algebras (BEQ-algebras) and investigate some properties. Then, we introduce a congruence relation on good BEQ-algebras by using ideals, and then, we solve an open problem in Paad (2019). Moreover, we show that in IEQ-algebras, there is a one-to-one correspondence between congruence relations and the set of ideals. In the following, we characterize the generated preideal in BEQ-algebras, and by using this, we prove that the family of all preideals of a BEQ-algebra is a complete lattice. Then, we show that the family of all preideals of a prelinear IEQ-algebras is a distributive lattice and becomes a Heyting algebra. Finally, we show that we can construct an MV-algebra from the family of all preideals of a prelinear IEQ-algebra.

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Notes

  1. Given an algebra \(<E,F>\), where F is a set of operations on E and \(F'\subseteq F\), then the algebra \(<E,F'>\) is called the \(F'\)-reduct of \(<E,F>\).

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Acknowledgements

This research is supported by a grant of National Natural Science Foundation of China (11971384).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

    N. Akhlaghinia & R. A. Borzooei

  2. Hatef Higher Education Institute, Zahedan, Iran

    M. Aaly Kologani

Authors
  1. N. Akhlaghinia
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  2. R. A. Borzooei
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  3. M. Aaly Kologani
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Correspondence to R. A. Borzooei.

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Akhlaghinia, N., Borzooei, R.A. & Kologani, M.A. Preideals in EQ-algebras. Soft Comput 25, 12703–12715 (2021). https://doi.org/10.1007/s00500-021-06071-y

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