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Green’s relations on the strong endomorphism monoid of a graph. (English) Zbl 0791.20077
An endomorphism of a graph is called strong if in addition to preserving edges it also reflects edges. The author describes inside the graph Green’s relations \(\mathcal L\), \(\mathcal R\), \(\mathcal H\) and \(\mathcal D\) for pairs of strong graph endomorphisms. As a consequence the author proves that the monoid of strong endomorphisms \(S\text{ End}(G)\) is regular. The methods are used to construct \(u \in S \text{ End}(G)\) with \(uf = g\) for \(f,g \in S \text{ End}(G)\) with \((f,g) \in {\mathcal L}\). All graphs are finite and undirected.

MSC:
20M20 Semigroups of transformations, relations, partitions, etc.
20M30 Representation of semigroups; actions of semigroups on sets
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20M15 Mappings of semigroups
20M17 Regular semigroups
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References:
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