On ideal theory of hoops | Zendy

M. Aaly Kologani | Zendy; R. A. Borzooei | Zendy

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On ideal theory of hoops

Author(s) -

M. Aaly Kologani,

R. A. Borzooei

Publication year - 2019

Publication title -

mathematica bohemica

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.254

H-Index - 11

eISSN - 2464-7136

pISSN - 0862-7959

DOI - 10.21136/mb.2019.0140-17

Subject(s) - mathematics , quotient , prime (order theory) , ideal (ethics) , maximal ideal , negation , pure mathematics , congruence (geometry) , relation (database) , property (philosophy) , simple (philosophy) , congruence relation , minimal ideal , boolean prime ideal theorem , algebra over a field , extension (predicate logic) , combinatorics , computer science , geometry , law , philosophy , epistemology , database , political science , programming language

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.

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