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Prefilters and fuzzy prefilters on pseudo EQ-algebras

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Abstract

In this paper, we mainly study prefilters and fuzzy prefilters on pseudo EQ-algebras, and give a series of properties of them. To begin with, we give some new results about prefilters on pseudo EQ-algebras and present that the set PF(\({\mathcal {E}}\)) of all prefilters on a good pseudo \(\ell\)EQ-algebra forms a Heyting algebra. In the following, we introduce three types of prefilters on pseudo EQ-algebras, which are implicative prefilters, positive implicative prefilters and fantastic prefilters, investigate their related properties, and discuss the relationships among them. We give some conditions under which positive implicative prefilters and implicative prefilters can be mutually induced and some conditions under which each implicative prefilter coincides with a fantastic prefilter. Moreover, we give the fuzzy versions of prefilters and these three special prefilters, investigate their related properties, and discuss the algebraic structures of the set of all fuzzy prefilters. It is proved that the set FPF(\({\mathcal {E}}\)) of all fuzzy prefilters on a residuated pseudo EQ-algebra forms a residuated lattice, and also a Heyting algebra. In the last, we investigate their relationships among fuzzy implicative prefilters, fuzzy positive implicative prefilters and fuzzy fantastic prefilters. We give some conditions under which fuzzy implicative prefilters are equivalent to fuzzy positive implicative prefilters, and some conditions under which a fuzzy implicative prefilter is coincident with a fuzzy fantastic prefilter.

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Funding

This work is supported by the National Natural Science Foundation of China (No. 12331016).

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  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, China

    Jieqiong Shi & Bin Zhao

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  1. Jieqiong Shi
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  2. Bin Zhao
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Correspondence to Bin Zhao.

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Shi, J., Zhao, B. Prefilters and fuzzy prefilters on pseudo EQ-algebras. Int. J. Mach. Learn. & Cyber. (2024). https://doi.org/10.1007/s13042-023-02042-x

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