Geodesic
On ideal theory of hoops
Mathematica Bohemica, Tome 145 (2020) no. 2, pp. 141-162
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we define and characterize the notions of (implicative, maximal, prime) ideals in hoops. Then we investigate the relation between them and prove that every maximal implicative ideal of a $\vee $-hoop with double negation property is a prime one. Also, we define a congruence relation on hoops by ideals and study the quotient that is made by it. This notion helps us to show that an ideal is maximal if and only if the quotient hoop is a simple MV-algebra. Also, we investigate the relationship between ideals and filters by exploiting the set of complements.
In this paper, we define and characterize the notions of (implicative, maximal, prime) ideals in hoops. Then we investigate the relation between them and prove that every maximal implicative ideal of a $\vee $-hoop with double negation property is a prime one. Also, we define a congruence relation on hoops by ideals and study the quotient that is made by it. This notion helps us to show that an ideal is maximal if and only if the quotient hoop is a simple MV-algebra. Also, we investigate the relationship between ideals and filters by exploiting the set of complements.
DOI : 10.21136/MB.2019.0140-17
Classification : 03G25, 06B99
Keywords: Hoop; (implicative, maximal, prime) ideal; MV-algebra; Boolean algebra 
@article{10_21136_MB_2019_0140_17,
     author = {Aaly Kologani, Mona and Borzooei, Rajab Ali},
     title = {On ideal theory of hoops},
     journal = {Mathematica Bohemica},
     pages = {141--162},
     year = {2020},
     volume = {145},
     number = {2},
     doi = {10.21136/MB.2019.0140-17},
     mrnumber = {4221826},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0140-17/}
}
TY  - JOUR
AU  - Aaly Kologani, Mona
AU  - Borzooei, Rajab Ali
TI  - On ideal theory of hoops
JO  - Mathematica Bohemica
PY  - 2020
SP  - 141
EP  - 162
VL  - 145
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0140-17/
DO  - 10.21136/MB.2019.0140-17
LA  - en
ID  - 10_21136_MB_2019_0140_17
ER  - 
%0 Journal Article
%A Aaly Kologani, Mona
%A Borzooei, Rajab Ali
%T On ideal theory of hoops
%J Mathematica Bohemica
%D 2020
%P 141-162
%V 145
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0140-17/
%R 10.21136/MB.2019.0140-17
%G en
%F 10_21136_MB_2019_0140_17
Aaly Kologani, Mona; Borzooei, Rajab Ali. On ideal theory of hoops. Mathematica Bohemica, Tome 145 (2020) no. 2, pp. 141-162. doi: 10.21136/MB.2019.0140-17

[1] Alavi, S. Z., Borzooei, R. A., Kologani, M. Aaly: Filter theory of pseudo hoop-algebras. Ital. J. Pure Appl. Math. 37 (2017), 619-632. | MR | JFM

[2] Saeid, A. Borumand, Motamed, S.: Some results in BL-algebras. Math. Log. Q. 55 (2009), 649-658. | DOI | MR | JFM

[3] Borzooei, R. A., Kologani, M. Aaly: Filter theory of hoop-algebras. J. Adv. Res. Pure Math. 6 (2014), 72-86. | DOI | MR

[4] Borzooei, R. A., Kologani, M. Aaly: Stabilizer topology of hoops. J. Alg. Structures and Their Appl. 1 (2014), 35-48.

[5] Bosbach, B.: Komplementäre Halbgruppen. Axiomatik und Arithmetik. Fundam. Math. 64 (1969), 257-287 German. | DOI | MR | JFM

[6] Bosbach, B.: Komplementäre Halbgruppen. Kongruenzen und Quotienten. Fundam. Math. 69 (1970), 1-14 German. | DOI | MR | JFM

[7] Nola, A. Di, Leuştean, L.: Compact representations of BL-algebras. Arch. Math. Logic 42 (2003), 737-761. | DOI | MR | JFM

[8] Esteva, F., Godo, L.: Monoidal t-norm based logic, towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124 (2001), 271-288. | DOI | MR | JFM

[9] Georgescu, G., Leuştean, L., Preoteasa, V.: Pseudo-hoops. J. Mult.-Val. Log. Soft Comput. 11 (2005), 153-184. | MR | JFM

[10] Hájek, P.: Mathematics of Fuzzy Logic. Trends in Logic-Studia Logica Library 4. Kluwer Academic Publishers, Dordrecht (1998). | DOI | MR | JFM

[11] Kondo, M., Dudek, W. A.: Filter theory of BL-algebras. Soft Comput. 12 (2008), 419-423. | DOI | JFM

[12] Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Logic Without Contraction. Japan Advanced Institute of Science and Technology (2001).

[13] Namdar, A., Borzooei, R. A., Saeid, A. Borumand, Kologani, M. Aaly: Some results in hoop algebras. J. Intell. Fuzzy Syst. 32 (2017), 1805-1813. | DOI | JFM

Cité par Sources :