| Summary: | In this paper, we introduce the notions of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mo>∈</mo> <mo>,</mo> <mo>∈</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-fuzzy positive implicative filters and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mo>∈</mo> <mo>,</mo> <mo>∈</mo> <mo>∨</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-fuzzy positive implicative filters in hoops and investigate their properties. We also define some equivalent definitions of them, and then we use the congruence relation on hoop defined in blue[Aaly Kologani, M.; Mohseni Takallo, M.; Kim, H.S. Fuzzy filters of hoops based on fuzzy points. <i>Mathematics.</i> <b>2019</b>, <i>7</i>, 430; doi:10.3390/math7050430] by using an <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mo>∈</mo> <mo>,</mo> <mo>∈</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-fuzzy filter in hoop. We show that the quotient structure of this relation is a Brouwerian semilattice.
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