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On the rank of semigroup of transformations with restricted partial range

Jin Jiulin (School of Mathematical Sciences, Guizhou Normal University, Guiyang, China), JLJIN@gznu.edu.cn

Let T(X) be the full transformation semigroup on a nonempty set X. For Ø ≠ Z ⊆ Y ⊆ X, let T(X,Y,Z) = {α ∈ T(X):Yα ⊆ Z}. It is not difficult to see that it is a generalized form of three well-known semigroups. This paper obtains an isomorphism theorem of T(X,Y,Z). In addition, when X is finite and Z ⊂ Y ⊂ X, the rank of the semigroup T(X,Y,Z) is calculated.

Keywords: transformation semigroup, rank, restricted range, isomorphism theorem