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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