The Smarandache vertices of the annihilation graphs of commutator posets and lattices with respect to an element and an ideal
| dc.contributor.advisor | Elham Mehdinezhad | |
| dc.contributor.author | Siame, Happy | |
| dc.date.accessioned | 2025-05-21T09:44:16Z | |
| dc.date.available | 2025-05-21T09:44:16Z | |
| dc.date.issued | 2024 | |
| dc.description.abstract | 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]. | |
| dc.identifier.uri | https://hdl.handle.net/10566/20426 | |
| dc.language.iso | en | |
| dc.publisher | University of the Western Cape | |
| dc.subject | Commutator poset | |
| dc.subject | commutator lattice | |
| dc.subject | lower set | |
| dc.subject | annihilation graph | |
| dc.subject | complete bipartite (r-partite) graph | |
| dc.title | The Smarandache vertices of the annihilation graphs of commutator posets and lattices with respect to an element and an ideal | |
| dc.type | Thesis |