{"id":{"repo_id":"western-cape","oai_identifier":"oai:uwcscholar.uwc.ac.za:10566/20888"},"canonical_url":"https://search.dev.ndltd.org/etd/western-cape/oai:uwcscholar.uwc.ac.za:10566/20888","repository":{"repo_id":"western-cape","name":"University of the Western Cape","base_url":"https://uwcscholar.uwc.ac.za:8443/server/oai/request"},"display":{"title":"The Smarandache vertices of the annihilation graphs of commutator posets and lattices with respect to an element and an ideal","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].","abstract_html":"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].","abstract_has_math":false,"creators":["Siame Happy"],"institution":"University of the Western Cape","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024","date_published":"2024","updated_at":"2026-07-24T06:00:32Z","subjects":["Commutator poset","commutator lattice","lower set","ideal [m-ideal]","annihilation graph"],"languages":[],"rights":[],"rights_urls":["https://uwcscholar.uwc.ac.za/bitstreams/4a69e281-933e-4ebf-b0ad-9abf40a0e5cd/download"],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Siame Happy"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2024"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of the Western Cape"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://hdl.handle.net/10566/20888"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Commutator poset","commutator lattice","lower set","ideal [m-ideal]","annihilation graph"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["https://uwcscholar.uwc.ac.za/bitstreams/4a69e281-933e-4ebf-b0ad-9abf40a0e5cd/download"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://uwcscholar.uwc.ac.za/bitstreams/678508bd-b8e0-452f-b159-222652cf8ab8/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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]."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["77225bd32d18b16b391cb0bb815e2f3d","bb9bdc0b3349e4284e09149f943790b4","7d32036539d016448848219f96f5bc8b"]},{"key":"dc:title","label":"Title","values":["The Smarandache vertices of the annihilation graphs of commutator posets and lattices with respect to an element and an ideal"]}]}],"canonical_facts":{"dc:creator":["Siame Happy"],"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:format.checksum.md5":["77225bd32d18b16b391cb0bb815e2f3d","bb9bdc0b3349e4284e09149f943790b4","7d32036539d016448848219f96f5bc8b"],"dc:identifier.uri":["https://uwcscholar.uwc.ac.za/bitstreams/678508bd-b8e0-452f-b159-222652cf8ab8/download"],"dc:publisher.institution":["University of the Western Cape"],"dc:relation.isreferencedby":["https://hdl.handle.net/10566/20888"],"dc:rights":["https://uwcscholar.uwc.ac.za/bitstreams/4a69e281-933e-4ebf-b0ad-9abf40a0e5cd/download"],"dc:subject":["Commutator poset","commutator lattice","lower set","ideal [m-ideal]","annihilation 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"]},"updated_at":"2026-07-24T06:00:32Z"}