On the Rank of the Semigroup TE(X) |
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Authors: | Pei Huisheng |
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Institution: | (1) Department of Mathematics, Xinyang Normal University, Xinyang, Henan 464000, China |
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Abstract: | ${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a
nontrivial equivalence $E$ on $X$, let
\
T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} .
\]
Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of
the topological space $X$ for which the collection of all
$E$-classes is a basis. In this paper, we first discuss the rank
of the homeomorphism group $G$, and then consider the rank of $T_E
(X)$ for a special case that the set $X$ is finite and that each
class of the equivalence $E$ has the same cardinality. Finally,
the rank of the closed selfmap semigroup $\Gamma(X)$ of the space
$X$ is observed. We conclude that the rank of $G$ is no more than
4, the rank of $T_E (X)$ is no more than 6 and the rank of
$\Gamma(X)$ is no more than 5. |
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Keywords: | |
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