This question relates to Realizing groups as automorphism groups of graphs.
Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
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Sign up to join this communityThis question relates to Realizing groups as automorphism groups of graphs.
Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
Hedrlín and Pultr proved that every monoid was the endomorphism monoid of a graph. See their paper "Symmetric relations (undirected graphs) with given semigroup" Monatsh. Math 69 (1965), eudml, DOI: 10.1007/BF01297617.
Trying to forestall another question along these lines let me add that every locally presentable category fully embeds into the category of graphs - see Adámek, Rosický Locally presentable and accessible categories 1994.
In fact, depending on set theories in which we work, every concrete category fully embeds into graphs - see Pultr, Trnková, Combinatorial, algebraic and topological representations of groups, semigroups and categories 1980.
For example: the category of metrizable spaces embeds into Graphs and in some set theories the category of Hausdorff topological spaces also embeds into Graphs.
Monoid is a category with a single object, hence a special case of the first result.