From $$Z^{\circ }$$-ideals to weak $$Z^{\circ }$$-ideals in MV-algebras

Abstract In this paper, the arranged element is introduced in MV -algebras, and its characteristics are studied. It is proved that an MV -algebra A is an MV -chain if and only if every nonzero element is an arranged element. In the continuation, weak $$Z^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> -ideals are defined and equivalent definitions of weak $$Z^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> -ideals are provided. Their relationships with existing notions such as $$Z^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> -ideals and $$Z^{\circ }_{J}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mi>J</mml:mi> <mml:mo>∘</mml:mo> </mml:msubsup> </mml:math> -ideals are examined. This analysis makes it possible to establish new classifications of prime and minimal prime ideals. Additionally, properties of annihilators in MV -algebras are investigated.