# Ideals in pseudo-hoop algebras

Document Type : Original Article

Authors

School of Mathematics and Statistics, Shandong Normal University, Jinan, P.R.China

Abstract

Pseudo-hoop algebras are non-commutative generalizations of hoop-algebras, originally introduced by Bosbach. In this paper, we study ideals in pseudo-hoop algebras. We define congruences induced by ideals and construct the quotient structure. We show that there is a one-toone correspondence between the set of all normal ideals of a pseudo-hoop algebra A with condition (pDN) and the set of all congruences on A. Also, we prove that if A is a good pseudo-hoop algebra with pre-linear condition, then a normal ideal P of A is prime if and only if A/P is a pseudo-hoop chain. Furthermore, we analyse the relationship between ideals and filters of A.

Keywords

#### References

[1] M. Aaly Kologani, R.A. Borzooei, On ideal theory of hoops, Mathematica Bohemica, 145(2)
(2020), 141–162.
[2] S.Z. Alavi, R.A. Borzooei, M. Aaly Kologani, Filter theory of pseud
[3] R.A. Borzooei, M. Aaly Kologani, Stabilizer topology of hoops, Algebraic Structures and Their
Applications, 1(1) (2014), 35–48.
[4] B. Bosbach, Komplementäre Halbgruppen. Axiomatik und Arithmetik, Fundamenta Mathematicae,
64 (1969), 257–287.
[5] B. Bosbach, Komplementäre Halbgruppen. Kongruenzen und Quotienten, Fundamenta Mathematicae,
69 (1970), 1–14.
[6] M. Botur, A. Dvurečenskij, On pseudo-BL-algebras and pseudo-hoops with normal maximal
filters, Soft Computing, 20(2) (2016), 439–448.
[7] M. Botur, A. Dvurečenskij, T. Kowalski, On normal-valued basic pseudo-hoops, Soft Computing,
16 (2012), 635–644.
[8] L. C. Ciungu, Non-commutative multiple-valued logic algebras, Springer, New York, 2014.
[9] L. C. Ciungu, Involutive filters of pseudo-hoops, Soft Computing, 23 (2019), 9459–9476.
[10] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: Part I, Multiple-Valued Logic,
8(5-6) (2002), 673–714.
[11] A. Dvurečenskij, States on pseudo MV-algebras, Studia logica, 68 (2001), 301–327.
[12] G. Georgescu, A. Iorgulescu, Pseudo-MV algebras, Multiple-Valued Logic, 6(1-2) (2001), 95–
135.
[13] G. Georgescu, L. Leuştean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued Logic
and Soft Computing, 11(1-2) (2005), 153–184.
[14] C. Lele, J.B. Nganou, MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems,
218 (2013), 103–113.
[15] A. Namdar, R.A. Borzooei, Nodal filters in hoop algebras, Soft Computing, 22 (2018), 7119–
7128.
[16] J. Rachunek, D. Šalounová, Ideals and involutive filters in generalizations of fuzzy structures,
Fuzzy Sets and Systems, 311 (2017), 70–85.