| Summary: | The aim of this paper is to classify all Hopf algebra structures on the quotient of Ore extensions <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">H</mi><mn>4</mn></msub><mrow><mo>[</mo><mi>z</mi><mo>;</mo><mi>σ</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula> of automorphism type for the Sweedler<inline-formula><math display="inline"><semantics><msup><mrow></mrow><mo>′</mo></msup></semantics></math></inline-formula>s 4-dimensional Hopf algebra <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">H</mi><mn>4</mn></msub></semantics></math></inline-formula>. Firstly, we calculate all equivalent classes of twisted homomorphisms <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>σ</mi><mo>,</mo><mi>J</mi><mo>)</mo></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">H</mi><mn>4</mn></msub></semantics></math></inline-formula>. Then we give the classification of all bialgebra (Hopf algebra) structures on the quotients of <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">H</mi><mn>4</mn></msub><mrow><mo>[</mo><mi>z</mi><mo>;</mo><mi>σ</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula> up to isomorphism.
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