Generalized Stonean BE-algebras

In this paper, the notion of a generalized Stonean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="italic">BE</mml:mi></mml:mrow></mml:math>-algebra is introduced. A set of equivalent conditions is given for every quasi-complemented <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="italic">BE</mml:mi></mml:mrow></mml:math>-algebra to become a generalized Stonean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="italic">BE</mml:mi></mml:mrow></mml:math>-algebra. A necessary and sufficient condition is stated for a self-distributive and commutative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="italic">BE</mml:mi></mml:mrow></mml:math>-algebra to become a generalized Stonean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="italic">BE</mml:mi></mml:mrow></mml:math>-algebra. The concept of Stonean filters is introduced and then generalized Stonean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="italic">BE</mml:mi></mml:mrow></mml:math>-algebras are characterized by Stonean filters. The notion of hyper Stonean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="italic">BE</mml:mi></mml:mrow></mml:math>-algebras is introduced and then a characterization theorem in terms of prime filters is given.