Quotient Hoops Induced by Quasi-Valuation Maps

R. A. Borzooei, G. R. Rezaei, M. Aaly Kologani, Y. B. Jun

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.