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The hull-kernel topology on prime filters in residuated lattices

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Abstract

The notion of the (dual) hull-kernel topology on a collection of prime filters in a residuated lattice is introduced and investigated. It is observed that any collection of prime filters is a \(T_0\) topological space under the (dual) hull-kernel topology. It is proved that any collection of prime filters is a \(T_1\) space if and only if it is an antichain, and it is a Hausdorff space if and only if it satisfies some certain conditions. Some characterizations in which maximal filters form a Hausdorff space are given. In the end, we focus on the space of minimal prim filters, and verify that this space is totally disconnected Hausdorff. This paper is closed by description of the compactness of the space of the minimal prime filters using the space of prime \(\alpha \)-filters.

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Acknowledgements

We would like to give heartfelt thanks to the referees for their very careful reading of the paper and for their very valuable comments and suggestions which improved the paper.

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  1. Department of Mathematics, Persian Gulf University, Bushehr, Iran

    Saeed Rasouli & Amin Dehghani

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  1. Saeed Rasouli
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  2. Amin Dehghani
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Correspondence to Saeed Rasouli.

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Rasouli, S., Dehghani, A. The hull-kernel topology on prime filters in residuated lattices. Soft Comput 25, 10519–10541 (2021). https://doi.org/10.1007/s00500-021-05985-x

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