The annihilation graphs of commutator posets and lattices with respect to an ideal

Abstract

As an extension of [Formula: see text] (the annihilation graph of the commutator poset (lattice) [Formula: see text] with respect to an element [Formula: see text]), we discuss when [Formula: see text] (the annihilation graph of the commutator poset (lattice) [Formula: see text] with respect to an ideal [Formula: see text]) is a complete bipartite ([Formula: see text]-partite) graph together with some of its other graph-theoretic properties. We investigate the interplay between some (order-) lattice-theoretic properties of [Formula: see text] and graph-theoretic properties of its associated graph [Formula: see text]. 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 [Formula: see text] is properly contained in the class of [Formula: see text]-ideals of [Formula: see text] (i.e. multiplicatively absorptive ideals (sets) of [Formula: see text] that are defined by commutator operation).