THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION
2013 ◽
Vol 88
(3)
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pp. 359-368
Keyword(s):
AbstractLet ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote $$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$ so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.
2013 ◽
Vol 12
(08)
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pp. 1350041
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2012 ◽
Vol 87
(1)
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pp. 94-107
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2008 ◽
Vol 78
(1)
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pp. 117-128
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1992 ◽
Vol 120
(1-2)
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pp. 129-142
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2013 ◽
Vol 13
(02)
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pp. 1350088
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1987 ◽
Vol 101
(3)
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pp. 395-403
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1998 ◽
Vol 57
(1)
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pp. 59-71
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