| Summary: | In this paper, we study the connection between generalized quasi-left alter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra and commutative Clifford semigroup by introducing the concept of an adjoint semigroup. We introduce QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula> algebra, in which every element is a quasi-minimal element, and prove that each QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula> algebra is equivalent to generalized quasi-left alter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra. Then, we introduce the notion of generalized quasi-left alter-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra and prove that every generalized quasi-left alter-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra is a generalized quasi-left alter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra. Next, we propose a new notion of quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula> algebra and discuss the relationship among them. Moreover, we study the subalgebras of quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula> algebra and the relationships between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>v</mi></msub></semantics></math></inline-formula>-group and quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra, hypergroup and quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra. Finally, we propose the concept of a generalized quasi-left alter quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula> algebra and QM-quasi hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra and discuss the relationships between them and related <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra.
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