Revistes Catalanes amb Accés Obert (RACO)

Molecules and linearly ordered ideals of $MV$-algebras

C. S. Hoo


We show that an ideal $I$ of an $MV$-algebra $A$ is linearly ordered if and only if every non-zero element of $I$ is a molecule. The set of molecules of $A$ is contained in $\operatorname{Inf}(A)\cup B_2(A)$ where $B_2(A)$ is the set of all elements $x\in A$ such that $2x$ is idempotent. It is shown that $I\ne \{0\}$ is weakly essential if and only if $B^\perp \subset B(A).$ Connections are shown among the classes of ideals that have various combinations of the properties of being implicative, essential, weakly essential, maximal or prime.

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