The Smarandache vertices of the annihilation graphs of commutator posets and lattices with respect to an element and an ideal
A vertex a in a simple graph G is said to be a Smarandache vertex (or S-vertex for short) provided that there exist three distinct vertices x, y, and b (all different from a) in G such that x—a, a—b, and b—y are edges in G, but there is no edge between x and y. In this interdisciplinary subject, we investigate the interplay between the algebraic properties of the commutator posets and lattices and their associated annihilation graphs with respect to an element [resp. an ideal] using the notion of the Smarandache vertices. Actually, AGz(L) (the annihilation graph of the commutator poset [lattice] L with respect to an element z ∈ L) [resp. AGI(L) (the annihilation graph of the commutator poset [lattice] L with respect to an ideal I ⊆ L, where AGI(L) is an extension of AGz(L) from an element to an ideal of L)] is a widely generalized context for the study of the zerodivisor type (annihilating-ideal) graphs, where the vertices of the graphs are not elements/ideals of a commutative ring, but elements of an abstract ordered set [lattice] (imitating the lattice of ideals of a ring), equipped with a commutative (not necessarily associative) binary operation (imitating the product of ideals of a ring). We discuss when AGz(L) [resp. AGI(L)] is a complete 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 graphtheoretic properties of its associated graph AGz(L) [resp. AGI(L)]. We provide some examples to show that some conditions are not superfluous assumptions. We prove and show by an example 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].
Commutator poset, commutator lattice, lower set, ideal [m-ideal], annihilation graph
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