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For each simple digraph D, we introduce its associated semigroup S(D) and its widened digraph W(D). We show that, for any two finite simple digraphs D1 and D2, S(D1) and S(D2) are isomorphic as semigroups if and only if W(D1) and W(D2) are isomorphic as digraphs.

Let T n be the full transformation semigroup on a finite set X n = {1, 2, . . . , n}. Let ρ be an equivalence relation on X n and be a total order on the partition set X n /ρ. We describe all automorphisms of the partition order-decreasing transformation monoid:that generalizes the results of Schreier (Fundam. Math., 28:261-264, 1936) and Šutov (Izv. Vysš. Učebn. Zaved., Mat., 3:177-184, 1961).

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