Geodesic
A note on Noetherian and Artinian hoops
Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 177-198.

Voir la notice de l'article provenant de la source Library of Science

The aim of this paper is defining the concepts of Noetherian and Artinian hoops by using the filter of hoop in the partial order set of all the filters of hoops and inclusion relation and find some equivalent definitions for this notion. We translate some important results from theory of rings to the case of hoop and their characterizations are established. The relation between short exact sequence on Noetherian and Artinian hoop studied and by using short exact sequence we prove that the Cartesian product of two hoops is Noetherian (Artinian) if and only if each one is a Noetherian (Artinian). By using the notion of filter in hoops, we define the notion of composition series and prove any ∨-hoop is Noetherian and Artinian if and only if it has composition series. Finally, Chinese Remainder theorem in hoop and the relation between maximal filter and Noetherian (Artinian) hoop are investigated.
Keywords: hoop, Noetherian hoop, Artinian hoop, filter, Chinese reminder, composition series 
@article{DMGAA_2024_44_1_a11,
     author = {Sabet Kish, Mehdi and Borzooei, Rajab Ali and Jabbari, Samad Haj and Aaly Kologani, Mona},
     title = {A note on {Noetherian} and {Artinian} hoops},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {177--198},
     publisher = {mathdoc},
     volume = {44},
     number = {1},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a11/}
}
TY  - JOUR
AU  - Sabet Kish, Mehdi
AU  - Borzooei, Rajab Ali
AU  - Jabbari, Samad Haj
AU  - Aaly Kologani, Mona
TI  - A note on Noetherian and Artinian hoops
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2024
SP  - 177
EP  - 198
VL  - 44
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a11/
LA  - en
ID  - DMGAA_2024_44_1_a11
ER  - 
%0 Journal Article
%A Sabet Kish, Mehdi
%A Borzooei, Rajab Ali
%A Jabbari, Samad Haj
%A Aaly Kologani, Mona
%T A note on Noetherian and Artinian hoops
%J Discussiones Mathematicae. General Algebra and Applications
%D 2024
%P 177-198
%V 44
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a11/
%G en
%F DMGAA_2024_44_1_a11
Sabet Kish, Mehdi; Borzooei, Rajab Ali; Jabbari, Samad Haj; Aaly Kologani, Mona. A note on Noetherian and Artinian hoops. Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 177-198. http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a11/

[1] M. Aaly Kologani and R.A. Borzooei, On ideal theory of hoop algebras, Math. Bohemica 145(2) (2020) 1–22. https://doi.org/10.21136/MB.2019.0140-17

[2] M. Aaly Kologani, Y.B. Jun, X.L. Xin, E.H. Roh and R.A. Borzooei, On co-annihilators in hoops, J. Int. and Fuzzy Syst. 37(4) (2019) 5471–5485. https://doi.org/10.3233/JIFS-190565

[3] M. Aaly Kologani, S.Z. Song, R.A. Borzooei and Y.B. Jun, Constructing some logical algebras with hoops, Mathematics 7(12) (2019) 1243. https://doi.org/10.3390/math7121243

[4] S.Z. Alavi, R.A. Borzooei and M. Aaly Kologani, Filter theory of pseudo hoop-algebras, Italian J. Pure Appl. Math. 37 (2017) 619–632.

[5] W.J. Blok and I.M.A. Ferreirim, Hoops and their implicational reducts, Alg. Meth. Logic and Comp. Sci. 28 (1993) 219–230.

[6] W.J. Blok and I.M.A. Ferreirim, On the structure of hoops, Algebra Univ. 43 (2000) 233–257. https://doi.org/10.1007/s000120050156

[7] R.A. Borzooei and M. Aaly Kologani, Filter theory of hoop-algebras, J. Adv. Res. Pure Math. 6 (2014) 72–86. https://doi.org/10.5373/jarpm

[8] R.A. Borzooei and M. Aaly Kologani, Results on hoops, J. Alg. Hyperstructures and Logical Alg. 1(1) (2020) 61–77. https://doi.org/10.29252/hatef.jahla.1.1.5

[9] R.A. Borzooei, M. Aaly Kologani, M. Sabet Kish and Y.B. Jun, Fuzzy positive implicative filters of hoops based on fuzzy points, Mathematics 7 (2019) 56. https://doi.org/10.3390/math7060566

[10] R.A. Borzooei, M. Sabetkish, E.H. Roh and M. Aaly Kologani, Int-soft filters in hoops, Int. J. Fuzzy Logic and Intel. Syst. 19(3) (2019) 213–222. https://doi.org/10.5391/IJFIS.2019.19.3.213

[11] B. Bosbach, Komplementäre halbgruppen. Axiomatik und arithmetik, Fundamenta Math. 64 (1969) 257–287.

[12] B. Bosbach, Komplementäre Halbgruppen. Kongruenzen und Quatienten, Fundamenta Math. 69 (1970) 1–14.

[13] D. Buşneag and D. Piciu, On the lattice of deductive systems of a BL-algebra, Central Eur. J. Math. 1(2) (2003) 221–238. https://doi.org/10.2478/BF02476010

[14] D. Buşneag, D. Piciu and A. Jeflea, Archimedean residuated lattices, Annals A.I. Cuza University, Mathematics 56 (2010) 227–252. https://doi.org/10.2478/v10157-010-0017-5

[15] R. Cignoli, F. Esteva, L. Godo and A. Torrens, Basic fuzzy logic is the logic of continuous t-norm and their residua, Soft Comp. 4 (2000) 106–112. https://doi.org/10.1007/s005000000044

[16] A. Di Nola, G. Georgescu and A. Iorgulescu, Pseudo-BL algebras: Part II, Multi-Valued Logic 8 (2002) 717–750.

[17] A. Dvureccenskij, A short note on categorical equivalences of proper weak pseudo EMV-algebras, J. Alg. Hyperstructures and Logical Alg. 3(1) (2022) 35–44. https://doi.org/10.52547/hatef.jahla.3.1.4

[18] R. Elohlavek, Some properties of residuated lattices, Czechoslovak Math. J. 53(123) (2003) 161–171.

[19] A. Filipoiu, G. Georgescu and A. Lettieri, Maximal MV-algebras, Mathware Soft Comp. 4(1) (1997) 53–62.

[20] T. Jeufack Yannick Lea, D. Joseph and T. Alomo Etienne Romuald, Residuated lattices derived from filters (ideals) in double Boolean algebras, J. Alg. Hyperstructures and Logical Alg. 3(2) (2022) 25–45. https://doi.org/10.52547/hatef.jahla.3.2.3

[21] S. Motamed and J. Moghaderi, Noetherian and Artinian BL-algebras, Soft Comp. 16 (2012) 1989–1994. https://doi.org/10.1007/s00500-012-0876-7

[22] F. Xie and H. Liu, Ideals in pseudo-hoop algebras, J. Alg. Hyperstructures and Logical Alg. 1(4) (2020) 39–53. https://doi.org/10.29252/hatef.jahla.1.4.3

[23] O. Zahiri, Chain conditions on BL-algebras, Soft Comp. 18 (2014) 419–426. https://doi.org/10.1007/s00500-013-1099-2