| Summary: | Let <i>X</i> be a nonempty set and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> the power set of <i>X</i>. The aim of this paper is to provide an explicit description of the monoid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>End</mi><msub><mn>1</mn><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msub></msub><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of unital ring endomorphisms of the Boolean ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the automorphism group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Aut</mi><mo>(</mo><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> when <i>X</i> is finite. Among other facts, it is shown that if <i>X</i> has cardinality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>End</mi><msub><mn>1</mn><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msub></msub><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≅</mo><msubsup><mi>T</mi><mi>n</mi><mrow><mi>o</mi><mi>p</mi></mrow></msubsup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>n</mi></msub></semantics></math></inline-formula> is the full transformation monoid on the set <i>X</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Aut</mi><mrow><mo>(</mo><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≅</mo><msub><mi>S</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>.
|
|---|