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The main purpose of the present paper is to introduced the notions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>∨</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><mi>S</mi><mi>A</mi><mi>s</mi></mrow></semantics></math></inline-formula> in subtraction BG-algebras. We provide different characterizations and some equivalent conditions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>∨</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><mi>S</mi><mi>A</mi><mi>s</mi></mrow></semantics></math></inline-formula> in terms of the level subsets of subtraction BG-algebras. It has been revealed that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo>,</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><mi>S</mi><mi>A</mi></mrow></semantics></math></inline-formula> are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><mi>S</mi><mi>A</mi></mrow></semantics></math></inline-formula> but the converse does not hold and an example is provided. We introduced <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>∨</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><mi>I</mi><mi>D</mi><mi>s</mi></mrow></semantics></math></inline-formula> and its some usual properties. In addition, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>h</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="fraktur">N</mi><mo stretchy="false">˜</mo></mover><mrow><mo stretchy="false">[</mo><mi>ς</mi><mo stretchy="false">]</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>∨</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><mi>I</mi><mi>D</mi></mrow></semantics></math></inline-formula>. Moreover, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>h</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="fraktur">N</mi><mo stretchy="false">˜</mo></mover><mrow><mo stretchy="false">[</mo><mi>ς</mi><mo stretchy="false">]</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>∨</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><mi>I</mi><mi>D</mi></mrow></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi mathvariant="fraktur">N</mi><mo stretchy="false">˜</mo></mover><mrow><mo stretchy="false">[</mo><mi>ς</mi><mo stretchy="false">]</mo></mrow></mrow></semantics></math></inline-formula> are an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>∨</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><mi>I</mi><mi>D</mi></mrow></semantics></math></inline-formula>. Finally, we characterize <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>∨</mo><msub><mover accent="true"><mi>q</mi><mo>´</mo></mover><mover accent="true"><mi>k</mi><mo>ˇ</mo></mover></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><msub><mi>H</mi><mrow><mi>I</mi><mi>D</mi></mrow></msub></mrow></semantics></math></inline-formula> which is a generalization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>,</mo><mover accent="true"><mo>∈</mo><mo>´</mo></mover><mo>∨</mo><mover accent="true"><mi>q</mi><mo>´</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><msub><mi>I</mi><mrow><mi>F</mi><mi>S</mi></mrow></msub><msub><mi>H</mi><mrow><mi>I</mi><mi>D</mi></mrow></msub></mrow></semantics></math></inline-formula>.