Geodesic
Quotient Hoops Induced by Quasi-Valuation Maps
Kragujevac Journal of Mathematics, Tome 46 (2022) no. 5, p. 743

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In this paper, our aim was making a metric space on hoop algebras, because of that, we introduced the notion of valuation maps from $F$-quasi-valuation map based on hoops and related properties of them are investigated. By using these notions, we introduced a quasi-metric space. The continuity of operations of a hoop is studied with topology induced by a quasi-valuation. Also, we studied hoop homomorphism and investigated that under which condition this homomorphism is an $F$-quasi-valuation map. Moreover, we wanted to find a congruence relation on hoops in a new way and study about the quotient structure that is made by it. So, we defined a congruence relation by $F$-quasi-valuation map and proved that the quotient is a hoop.
Classification : 06B30 03G25, 06D20
Keywords: Hoop, quasi-valuation map, $S_\odot$-quasi-valuation map, $S_\to$-quasi-valuation map, $F$-quasi-valuation map, (pseudo) metric space 
@article{KJM_2022_46_5_a6,
     author = {R. A. Borzooei and G. R. Rezaei and M. Aaly Kologani and Y. B. Jun},
     title = {Quotient {Hoops} {Induced} by {Quasi-Valuation} {Maps}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {743 },
     publisher = {mathdoc},
     volume = {46},
     number = {5},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2022_46_5_a6/}
}
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R. A. Borzooei; G. R. Rezaei; M. Aaly Kologani; Y. B. Jun. Quotient Hoops Induced by Quasi-Valuation Maps. Kragujevac Journal of Mathematics, Tome 46 (2022) no. 5, p. 743 . http://geodesic.mathdoc.fr/item/KJM_2022_46_5_a6/