Abstract
Let T(X) be the full transformation semigroup on a nonempty set X. For ? ? Z ? Y ? X, let T(X,Y,Z) = {? ? T(X):Y? ? 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.
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