On co-annihilators in hoops

• Published in: Journal of intelligent & fuzzy systems Vol. 37; no. 4; pp. 5471 - 5485
• Main Authors: Aaly Kologani, M., Jun, Y.B., Xin, X.L., Roh, E.H., Borzooei, R.A.
• Format: Journal Article
• Language: English
• Published: Amsterdam IOS Press BV 01.01.2019
• Subjects: Algebra; Boolean algebra; Hoops; Lattices
• ISSN: 1064-1246; 1875-8967
• DOI: 10.3233/JIFS-190565

In this paper, we introduce the notion of co-annihilator in hoops and investigate some related properties of them. Then we prove that the set of filters F ( A ) form two pseudo-complemented lattices (with ∗ and ⊤) that if A has (DNP), then the two pseudo-complemented lattices are the same. Moreover, by defining the operation → on the lattice F ( A ) , we prove that F ( A ) is a Heyting algebra and by defining of the product operation, we show that F ( A ) is a bounded hoop. Finally, we define the C - Ann (A) to be the set of all co-annihilators of A, then we have that it had made a Boolean algebra. Also, we give an extension of a filter, which leads to a useful characterization of α-filters on hoops. For instance, we obtain a series of characterizations of α-filters. In addition, we show that there are no non-trivial α-filters on hoop-chains. That implies the structure of all α-filters contains only trivial α-filters on hoops. On hoops, we prove that the set of all α-filters is a pseudo-complemented lattice. Moreover, the structure of all α-filters can form a Boolean algebra under certain conditions.