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As an extension of AGz(L) (the annihilation graph of the commutator poset
[lattice] L with respect to an element z â L), we discuss when AGI (L) (the annihilation
graph of the commutator poset [lattice] L with respect to an ideal I â L) is a complete
bipartite [r-partite] graph together with some of its other graph-theoretic properties.
We investigate the interplay between some (order-) lattice-theoretic properties of L and
graph-theoretic properties of its associated graph AGI (L). We provide some examples
to show that some conditions are not superfluous assumptions. We prove and show by a
counterexample that the class of lower sets of a commutator poset L is properly contained
in the class of m-ideals of L [i.e. multiplicatively absorptive ideals (sets) of L that are
defined by commutator operation].
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