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The present note is closely related to the papers [1] and [2] by the authors. As the results of them are used here, we preserve the definitions and notation. Our aim is to prove the following theorem:Let S 1 be a semigroup with a unity element, cardinality of S i beeing less than the first inaccessible cardinal. Then there exists a symmetric relation R on a set X such that G(R, X) under composition is isomorphic with S 1.Using "graph" language the theorem asserts that there is an lmdirected graph with the given semigroup S i of all endomorphisms.