Nodal filters in hoop algebras

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 t...

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Bibliographic Details
Published inSoft computing (Berlin, Germany) Vol. 22; no. 21; pp. 7119 - 7128
Main Authors Namdar, A., Borzooei, R. A.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2018
Springer Nature B.V
Subjects
Algebra
Artificial Intelligence
Computational Intelligence
Control
Engineering
Foundations
Hoops
Mathematical Logic and Foundations
Mechatronics
Monoids
Robotics
Heyting algebra
Node
Hertz algebra
Hilbert algebra
BCK-algebra
Kleene algebra
Semi-De Morgan algebra
Hoop
Nodal filter
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Summary: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 NF ( A ) , the set of all nodal filters in hoop A , we show that 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.
ISSN:1432-7643
1433-7479
DOI:10.1007/s00500-017-2986-8