New Branch of Intuitionistic Fuzzification in Algebras with Their Applications

The intuitionistic fuzzification in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi mathvariant="normal">ρ</mml:mi><mml:mo>-</mml:mo></mml:math>algebras about the concepts of ideals and subalgebras given with several related characterizations is considered. Some new concepts like intuitionistic fuzzy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi mathvariant="normal">ρ</mml:mi><mml:mo>-</mml:mo></mml:math>ideal (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>I</mml:mi><mml:mi>F</mml:mi><mml:mi mathvariant="normal">ρ</mml:mi><mml:mi>i</mml:mi></mml:math>), intuitionistic fuzzy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi mathvariant="normal">ρ</mml:mi><mml:mo>-</mml:mo></mml:math>subalgebra (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>I</mml:mi><mml:mi>F</mml:mi><mml:mi mathvariant="normal">ρ</mml:mi><mml:mi>s</mml:mi></mml:math>), <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi mathvariant="normal">ρ</mml:mi><mml:mo>-</mml:mo></mml:math>homomorphism, and intuitionistic fuzzy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mover accent="false"><mml:mrow><mml:mi mathvariant="normal">ρ</mml:mi></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>-</mml:mo></mml:math>ideal (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mi>I</mml:mi><mml:mi>F</mml:mi><mml:mover accent="false"><mml:mrow><mml:mi mathvariant="normal">ρ</mml:mi></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:math>) are introduced and some of their descriptions are given in this work. Further, we show some applications on the family of all intuitionistic fuzzy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mi mathvariant="normal">ρ</mml:mi><mml:mo>-</mml:mo></mml:math>subalgebras <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>I</mml:mi><mml:mi>F</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant="normal">ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="fraktur">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mi mathvariant="normal">ρ</mml:mi><mml:mo>-</mml:mo></mml:math>algebra like the binary relations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mrow><mml:munder><mml:mrow><mml:mo>≈</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:munder></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mrow><mml:munder><mml:mrow><mml:mo>≈</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">ν</mml:mi></mml:mrow></mml:munder></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M15"><mml:mi>I</mml:mi><mml:mi>F</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant="normal">ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="fraktur">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. Also, their equivalence classes are given and studied.