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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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