Mathematica Bohemica, Vol. 145, No. 2, pp. 141-162, 2020


On ideal theory of hoops

Mona Aaly Kologani, Rajab Ali Borzooei

Received December 18, 2017.   Published online March 26, 2019.

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.
Keywords:  Hoop; (implicative, maximal, prime) ideal; MV-algebra; Boolean algebra
Classification MSC:  06B99, 03G25


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Affiliations:   Mona Aaly Kologani, Hatef Higher Education Institute, Bozorgmehr, Zahedan, Iran, e-mail: mona4011@gmail.com; Rajab Ali Borzooei, Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, District 1, Daneshjou Boulevard, 19839 69411, Tehran, Iran, e-mail: borzooei@sbu.ac.ir


 
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