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On weak implication algebra

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Abstract

In this paper, we introduce the notion of implication filter as a generalization of Boolean filter of first kind in Hilbert algebra and study it in detail. Also, we prove that F is an implication filter of a Hilbert algebra H if and only if every implicative filter of quotient algebra H / F is an implication filter. Finally, we generalized Boolean algebra and introduced a weak implication algebra, we prove that F is an implication filter of a Hilbert algebra H if and only if the quotient algebra H / F is a weak implication algebra. By suitable diagrams, we summarize the results of this paper and the previous results in these fields.

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Acknowledgements

We would like to warmly thank the referees for their helpful comments and suggestions for the improvement of this paper.

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Authors and Affiliations

  1. Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran

    Arsham Borumand Saeid

  2. Shahid Chamran Faculty, Technical and Vocational University, Kerman, Iran

    Ali Soleimani Nasab

  3. Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

    Ali Soleimani Nasab & Arsham Borumand Saeid

Authors
  1. Ali Soleimani Nasab
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  2. Arsham Borumand Saeid
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Correspondence to Arsham Borumand Saeid.

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Communicated by A. Di Nola.

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Soleimani Nasab, A., Borumand Saeid, A. On weak implication algebra. Soft Comput 23, 5393–5400 (2019). https://doi.org/10.1007/s00500-018-3495-0

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