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In this paper, we introduce the notion of a <inline-formula><math display="inline"><semantics><mrow><mi>B</mi><mi>V</mi></mrow></semantics></math></inline-formula>-algebra, and we show that a <inline-formula><math display="inline"><semantics><mrow><mi>B</mi><mi>V</mi></mrow></semantics></math></inline-formula>-algebra is logically equivalent to several algebras, i.e., <inline-formula><math display="inline"><semantics><mrow><mi>B</mi><mi>M</mi></mrow></semantics></math></inline-formula>-algebras, <inline-formula><math display="inline"><semantics><mrow><mi>B</mi><mi>T</mi></mrow></semantics></math></inline-formula>-algebras, <inline-formula><math display="inline"><semantics><mrow><mi>B</mi><mi>O</mi></mrow></semantics></math></inline-formula>-algebras and 0-commutative <i>B</i>-algebras. Moreover, we show that a <inline-formula><math display="inline"><semantics><mrow><mi>B</mi><mi>V</mi></mrow></semantics></math></inline-formula>-algebra with <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula> is logically equivalent to several algebras, and we show some relationships between a <inline-formula><math display="inline"><semantics><mrow><mi>B</mi><mi>V</mi></mrow></semantics></math></inline-formula>-algebra with <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula> and several related algebras.