Abstract
Let Tn be the full transformation semigroup on a finite set Xn={1,2,…,n}. Let ρ be an equivalence relation on Xn and ⪯ be a total order on the partition set Xn/ρ. We describe all automorphisms of the partition order-decreasing transformation monoid: $$T(\rho,\preceq)=\bigl\{\alpha\in T_{n}: (x\alpha)\rho\preceq x\rho, \forall x\in X_{n}\bigr\} $$ that generalizes the results of Schreier (Fundam. Math., 28:261–264, 1936) and Sutov (Izv. Vyss. Ucebn. Zaved., Mat., 3:177–184, 1961).
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