The annihilation graphs of commutator posets and lattices with respect to an element
| Autor: | Elham Mehdi-Nezhad, Amir M. Rahimi |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Algebra and Number Theory
Applied Mathematics 010102 general mathematics Commutator (electric) 0102 computer and information sciences Commutative ring Congruence relation 01 natural sciences Complete bipartite graph law.invention Combinatorics 010201 computation theory & mathematics law Binary operation Lattice (order) 0101 mathematics Partially ordered set Commutative property Mathematics |
| Zdroj: | Journal of Algebra and Its Applications. 16:1750106 |
| ISSN: | 1793-6829 0219-4988 |
| DOI: | 10.1142/s0219498817501067 |
| Popis: | We propose a new, widely generalized context for the study of the zero-divisor type (annihilating-ideal) graphs, where the vertices of 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 [Formula: see text] (the annihilation graph of the commutator poset [lattice] [Formula: see text] with respect to an element [Formula: see text]) is a complete bipartite graph together with some of its other graph-theoretic properties. In contrast to the case of rings, we construct a commutator poset whose [Formula: see text] contains a cut-point. We provide some examples to show that some conditions are not superfluous assumptions. We also give some examples of a large class of lattices, such as the lattice of ideals of a commutative ring, the lattice of normal subgroups of a group, and the lattice of all congruences on an algebra in a variety (congruence modular variety) by using the commutators as the multiplicative binary operation on these lattices. This shows that how the commutator theory can define and unify many zero-divisor type graphs of different algebraic structures as a special case of this paper. |
| Databáze: | OpenAIRE |
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