The monoid of the random graph
type: Item
instance of: scholarly article
title: The monoid of the random graph
zbMATH Open document ID: 0957.20045
DOI: 10.1007/PL00006009
author: Anthony Bonato
author: Dejan Delić
published in: Semigroup Forum
publication date: 2001-03-21T00:00:00Z
review text: The countable random graph \(R\) is the unique graph with the property that given any two non-empty disjoint finite sets of vertices there is a vertex in neither set that is adjacent to all members of the first set but to no member of the second. This paper has as its subject \(\text{End}(R)\), the monoid of all endomorphisms of \(R\), where a morphism between graphs is a mapping that preserves adjacency. It is proved that \(\text{End}(R)\) has no zero, is not regular, nor is it idempotent-generated. However, the cardinality of the set of minimal idempotents is that of the continuum and every countable linear order is embedded in the poset of idempotents of \(\text{End}(R)\).
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 05C80
Mathematics Subject Classification ID: 06A05
zbMATH DE Number: 1491105
zbMATH Keywords: monoids of endomorphisms
zbMATH Keywords: countable random graphs
zbMATH Keywords: minimal idempotents
zbMATH Keywords: countable linear orders
zbMATH Keywords: posets of idempotents
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
full work available at URL: pl00006009
OpenAlex ID: W2019621810
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description: scientific article; zbMATH DE number 1491105