Abstract
Hoop algebras or hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach (Fundam Math 64:257–287, 1969, Fundam Math 69:1–14, 1970). In this paper, we introduce the notions of node and nodal filter in hoops and study some properties of them. First, we prove that the sets of all nodes are a bounded distributive lattice. Then by define some operations on \({\mathcal {NF}}(A)\), the set of all nodal filters in hoop A, we show that \({\mathcal {NF}}(A)\) is a Hertz algebra, Heyting algebra, Kleene algebra, semi-De Morgan algebra, Hilbert algebra and BCK-algebra. Finally, we investigate the relation among nodal filters and (positive) implicative, obstinate, prime and maximal filters in any hoops.
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Acknowledgements
The authors wish to express their appreciation for several excellent suggestions for improvements in this paper made by the editor and referees. Funding was provided by Shahid Beheshti University and Islamic Azad University Kerman.
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Communicated by A. Di Nola.
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Namdar, A., Borzooei, R.A. Nodal filters in hoop algebras. Soft Comput 22, 7119–7128 (2018). https://doi.org/10.1007/s00500-017-2986-8
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DOI: https://doi.org/10.1007/s00500-017-2986-8