### Quotient Hoops Induced by Quasi-Valuation Maps

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**Authors:**R. A. BORZOOEI, G. R. REZAEI, M. AALY KOLOGANI AND Y. B. JUN

**DOI:**10.46793/KgJMat2205.743B

**Abstract:**

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.

**Keywords:**

Hoop, quasi-valuation map, S

_{⊙}-quasi-valuation map, S

_{→}-quasi-valuation map, F-quasi-valuation map, (pseudo) metric space.

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