Maximal and essential ideas of MV-algebras
C. S. Hoo · RACO (Revistes Catalanes amb Accés Obert) (Consorci de Serveis Universitaris de Catalunya) · 1995
We show that an atom free ideal is densely ordered. It is shown that if $I$ is a maximal ideal of an $MV\\mbox{-algebra}\\;A,$ then =I^\\perp\\oplus I^{\\perp\\perp}$ where $I^\\perp=\\{x\\vert x\\le e\\}$ and $I^{\\perp\\perp} =\\{x\\vert x\\le \\bar e\\}$ for a unique idempotent $e.$ The socle, radical and implicative radical \t\t\t\t of $A$ are computed in certain cases. It is shown that if $A$ is not atom free but $I$ is a maximal ideal which is atom free, then $I$ is densely ordered, and $I=\\la At(A)\\ra^\\perp=\\la a\\ra^\\perp$ where $At(A)$ is the set of atoms of $A$ and $a\\in At(A).$ Then $A=I^\\perp\\oplus I^{\\perp\\perp}$ where $I^\\perp$ is atomic and $I^{\\perp\\perp}$ is atom free.