Affiliations: [a]
Hatef Higher Education Institute, Zahedan, Iran
| [b] Department of Mathematics Education, Gyeongsang National University, Korea
| [c] School of Mathematics, Northwest University, China
| [d] Department of Mathematics Education, Chinju National University of Education, Korea
| [e] Department of Mathematics, Shahid Beheshti University, Tehran, Iran
Correspondence:
[*]
Corresponding author. R.A. Borzooei, School of Mathematics, Northwest University, Xi’an, 710127, China. E-mail: xlxin@nwu.edu.cn.
Abstract: 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.
