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THE NATURAL PARTIAL ORDER ON SOME TRANSFORMATION SEMIGROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 2 / April 2014
- Published online by Cambridge University Press:
- 28 June 2013, pp. 279-292
- Print publication:
- April 2014
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- Article
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For a semigroup
$S$, let 
${S}^{1} $ be the semigroup obtained from 
$S$ by adding a new symbol 1 as its identity if 
$S$ has no identity; otherwise let 
${S}^{1} = S$. Mitsch defined the natural partial order 
$\leqslant $ on a semigroup 
$S$ as follows: for 
$a, b\in S$, 
$a\leqslant b$ if and only if 
$a= xb= by$ and 
$a= ay$ for some 
$x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.
THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
- Published online by Cambridge University Press:
- 25 January 2013, pp. 359-368
- Print publication:
- December 2013
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Let
${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set 
$X$ and 
$E$ be a nontrivial equivalence relation on 
$X$. Denote so that
$$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$
${T}_{\exists } (X)$ 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.
