Semigroup Forum

, Volume 85, Issue 3, pp 513-524

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Automorphisms of partition order-decreasing transformation monoids

  • Haobo YangAffiliated withQiangjiang College, Hangzhou Normal University
  • , Xiuliang YangAffiliated withDepartment of Mathematics, Hangzhou Normal University Email author 


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:
$$T(\rho,\preceq)=\bigl\{\alpha\in T_{n}: (x\alpha)\rho\preceq x\rho, \forall x\in X_{n}\bigr\} $$
that generalizes the results of Schreier (Fundam. Math., 28:261–264, 1936) and Šutov (Izv. Vysš. Učebn. Zaved., Mat., 3:177–184, 1961).


Transformation semigroup Partition order-decreasing Automorphism group