On ideal theory of hoops

Mona Aaly Kologani; Rajab Ali Borzooei

Mathematica Bohemica (2020)

  • Volume: 145, Issue: 2, page 141-162
  • ISSN: 0862-7959

Abstract

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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 ∨ -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.

How to cite

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Aaly Kologani, Mona, and Borzooei, Rajab Ali. "On ideal theory of hoops." Mathematica Bohemica 145.2 (2020): 141-162. <http://eudml.org/doc/296985>.

@article{AalyKologani2020,
abstract = {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.},
author = {Aaly Kologani, Mona, Borzooei, Rajab Ali},
journal = {Mathematica Bohemica},
keywords = {Hoop; (implicative; maximal; prime) ideal; MV-algebra; Boolean algebra},
language = {eng},
number = {2},
pages = {141-162},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On ideal theory of hoops},
url = {http://eudml.org/doc/296985},
volume = {145},
year = {2020},
}

TY - JOUR
AU - Aaly Kologani, Mona
AU - Borzooei, Rajab Ali
TI - On ideal theory of hoops
JO - Mathematica Bohemica
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 145
IS - 2
SP - 141
EP - 162
AB - 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.
LA - eng
KW - Hoop; (implicative; maximal; prime) ideal; MV-algebra; Boolean algebra
UR - http://eudml.org/doc/296985
ER -

References

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  4. Borzooei, R. A., Kologani, M. Aaly, Stabilizer topology of hoops, J. Alg. Structures and Their Appl. 1 (2014), 35-48. (2014) 
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  10. Hájek, P., 10.1007/978-94-011-5300-3, Trends in Logic-Studia Logica Library 4. Kluwer Academic Publishers, Dordrecht (1998). (1998) Zbl0937.03030MR1900263DOI10.1007/978-94-011-5300-3
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  12. Kowalski, T., Ono, H., Residuated Lattices: An Algebraic Glimpse at Logic Without Contraction, Japan Advanced Institute of Science and Technology (2001). (2001) 
  13. Namdar, A., Borzooei, R. A., Saeid, A. Borumand, Kologani, M. Aaly, 10.3233/JIFS-152553, J. Intell. Fuzzy Syst. 32 (2017), 1805-1813. (2017) Zbl1375.06015DOI10.3233/JIFS-152553

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