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Cubic multipolar structure with finite degree (briefly, cubic <i>k</i>-polar (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula>) structure) is a new hybrid extension of both <i>k</i>-polar fuzzy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>k</mi></msub><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula>) structure and cubic structure in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> structure consists of two parts; the first one is an interval-valued <i>k</i>-polar fuzzy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><msub><mi>V</mi><mi>k</mi></msub><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula>) structure acting as a membership grade extended from the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">P</mi><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">P</mi><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mi>k</mi></msup></mrow></semantics></math></inline-formula> (i.e., from interval-valued of real numbers to the <i>k</i>-tuple interval-valued of real numbers), and the second one is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>k</mi></msub><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula> structure acting as a nonmembership grade extended from the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mi>k</mi></msup></semantics></math></inline-formula> (i.e., from real numbers to the <i>k</i>-tuple of real numbers). This approach is based on generalized cubic algebraic structures using polarity concepts and therefore the novelty of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> algebraic structure lies in its large range comparative to both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>k</mi></msub><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula> algebraic structure and cubic algebraic structure. The aim of this manuscript is to apply the theory of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> structure on BCK/BCI-algebras. We originate the concepts of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> subalgebras and (closed) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> ideals. Moreover, some illustrative examples and dominant properties of these concepts are studied in detail. Characterizations of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> subalgebra/ideal are given, and the correspondence between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> subalgebras and (closed) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> ideals are discussed. In this regard, we provide a condition for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> subalgebra to be a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> ideal in a BCK-algebra. In a BCI-algebra, we provide conditions for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> subalgebra to be a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> ideal, and conditions for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> subalgebra to be a closed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> ideal. We prove that, in weakly BCK-algebra, every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> ideal is a closed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> ideal. Finally, we establish the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> extension property for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> ideal.