| Summary: | By considering an entry (i.e., a number, an idea, an object, etc.) which is represented by a known part <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and an unknown part <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>b</mi> <mi>T</mi> <mo>,</mo> <mi>c</mi> <mi>I</mi> <mo>,</mo> <mi>d</mi> <mi>F</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> <i>F</i> have their usual neutrosophic logic meanings and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> </mrow> </semantics> </math> </inline-formula> are real or complex numbers, Smarandache introduced the concept of neutrosophic quadruple numbers. Using the concept of neutrosophic quadruple numbers based on a set, Jun et al. constructed neutrosophic quadruple BCK/BCI-algebras and implicative neutrosophic quadruple BCK-algebras. The notion of a neutrosophic quadruple BCI-positive implicative ideal is introduced, and several properties are dealt with in this article. We establish the relationship between neutrosophic quadruple ideal and neutrosophic quadruple BCI-positive implicative ideal. Given nonempty subsets <i>I</i> and <i>J</i> of a BCI-algebra, conditions for the neutrosophic quadruple <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mo>,</mo> <mi>J</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-set to be a neutrosophic quadruple BCI-positive implicative ideal are provided.
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